9: Triangles
9.1 Measuring with Triangles
It is fair to ask why we should bother learning about triangles, since their relevance to ecology and natural resources isn’t immediately ap- parent. Indeed, natural materials tend to approximate more tabular or rounded shapes, and true triangles are harder to find in nature by comparison. But the real power of triangles is not in where we can see them, but where we can imagine them. Believe it or not, imaginary triangles can help us measure properties of a landscape or organism, and that fact is firmly incorporated into many of the tools and technologies that researchers and professionals use. In particular, the ratios between the lengths of triangle sides is one of their key as- sets. In this chapter, we’ll review some of the properties of triangles and see how these properties can be leveraged to measure things we care about.
9.2 Trigonometry primer
Trigonometry is the study of triangles, specifically the relationships between the lengths of their sides and the angles between them. At first glance, that might not seem like it is very relevant to the natural sciences, but a few examples might convince you otherwise:
- Determining the distance “as the crow flies” between two geographic points is often most easily done with the help of triangles.
- Measuring the height of a tree or a mountain can employ triangles.
- Telemetry often uses triangulation to determine the geographic location of collared animals.
If you’ve had a trigonometry class, you might associate the discipline with manipulating equations with sec 2 θ and cot(1 + π /2). Outside of math class, did you ever find yourself needing to find the secant of an angle? Not likely. But it isn’t too uncommon to encounter the likes of sine and cosine, which are often written sin and cos, respectively. That’s because they are really, really useful\(^{1}\). And it turns out, nearly all the other trig functions you may have learned about are readily defined using sines and cosines ! For example, the tangent of an angle can be defined as the ratio of the angle’s sine to its cosine, but it is so useful that we should recognize it as well.
As far as I’m concerned, on the off chance that you ever need to manipulate an equation containing the hyperbolic cosine (cosh) of something, you can look it up or type it into an internet tool like Wolfram|Alpha . If this is your first experience with trigonometry, have no fear, we’ll take it slow!
\(^{1}\)At this point you may just need to take my word for it, but I hope that you’ll appreciate this fact by the time you finish this chapter.
Wolfram|Alpha is a web-based tool developed by mathematician and entrepreneur Stephen Wolfram. It is based on the same underlying computational engine as the math software Mathematica, but can be used (for free with slightly-limited functionality) from any web browser. In addition to performing computation and algebraic simplification, it can attempt to comprehend simple written questions and can retrieve data from a few established databases, concerning for example weather, fincance, and sports.
Similarity
Similarity is a concept that may not boast enough sophistication to warrant its inclusion in a trigonometry class. However, it is an intuitively easy idea to grasp and its utility can be great. And fortunately for modern scientists, the formal application of similarity allows us to design tools for measuring things efficiently in the field.
The principle of geometric similarity\(^{2}\) is straight-forward. If we have a given shape with known side lengths and/or known angles formed between the sides, we can say that another shape is similar if it has the same number of sides and a relationship between those sides and angles that is the same as our reference shape. The two shapes can still be similar even if they are not the same size or orientation. If any combination of translation (moving the shape), rotation, reflection (a mirror image), or isometric\(^{3}\) scaling could allow you to overlay one shape on the other to find them to be identical, the shapes are similar.
In some fields this concept is endowed with a more sophisticated sounding name: geometric simili tude .
The word isometric in this context means that any change in one spatial dimension of a shape (e.g., length) is matched by a proportional change in all other dimensions.
For triangles, the qualifying criteria for similarity are simple, since there are only three sides and one internal angle at each of the three vertices. For shorthand, when two triangles have one angle that is identical, we’ll call that A . When two triangles have all three angles equal, we’ll refer to that is AAA . Likewise, if the side-length of one side of two triangles is equal, we’ll describe that with S . With these definitions, we’ll make the following claims, as yet unproven, about the criteria for determining similarity:
Two triangles are similar if any one of the following can be established:
• AAA . The angles of one triangle are equal to the angles of the second.
• SSS . The side-lengths of one triangle are equal to those of the second. Side lengths may be scaled by a constant C if that constant is the same for each side.
• SAS . Two side-lengths and one angle of one triangle are equal to those of the second. Side lengths may be scaled by a constant C if that constant is the same for each side.
When you’ve reached the end of this chapter, you should be able to show how each of these criteria for similarity could be derived from one of the others. That is left as an exercise for you to work on, one that can build your intuition for using triangle properties for problem solving.
9.2.1 The right triangle and sohcahtoa
The mnemonic SOH-CAH-TOA is one of a small number of things that most trig students remember years after taking the class. Indeed this is really helpful way to remember the algorithms relating the basic trig functions to ratios between the sides of a right triangle \(^{4}\). But it reveals nothing about the ways that triangles can be employed for practical purposes. So before we deal with these functions, let’s revisit what a right triangle is, where we might encounter one, and a few terms and rules regarding these beasts. Figure 9.2 shows a nice, well-behaved one.
As you may know, a right triangle is defined as a triangle with one right, or 90◦ angle
Notice that each vertex (a corner point) joins two of the three sides and doesn’t touch the side that is opposite it. For reasons that might be apparent in a moment, we choose names for the sides and vertices that imply a relationship between a vertex and the side opposite it. So for example, side a is opposite vertex A (meaning the vertex A is not one of the endpoints of side a ). It probably satisfies your intuition that the size of the angle at a vertex might have some simple relationships to the length of its opposite side – at least there is a more intuitive relationship there than between vertex A and one of the other two sides. Imagine keeping vertices A and C stationary,vbut allowing the angle at A to grow. It is plain to see that if the angle ∠ A increases, the vertex B must move up and the length of side a increases accordingly. This thought experiment produces similar results for the other opposing side/angle pairs as well, and we’ll use it to our advantage shortly in dealing with triangulation.
One of the most fundamental properties of all triangles is that the three vertex angles sum to 180\(^{◦}\) (∠ A + ∠ B + ∠ C = 180◦). For the special case of a right triangle, the right angle by definition is 90\(^{◦}\), so the other two angles must be smaller than 90 . This seems obvious, but it has an important consequence: the longest side of a triangle is the one that is opposite the largest angle. Therefore, since the right angle is the largest angle in a right triangle, the longest side (which we call the hypotenuse ) is across from the right angle (Figure 9.3).
In addition to the rule that angles must sum to 180◦, one of the most powerful properties of right triangles is their adherence to the Pythagorean theorem :
\[a^{2} +b^{2} = c^{2} \label{9.1} \]
This is always true provided that c is the hypotenuse of a right triangle. It turns out that there is a simple modification that can be made to this if we are dealing with any arbitrary triangle. But before we go much farther, consider an example setting where a right triangle can be a useful aid to measurement.
9.2.2 Example: overland distances
If you are visiting GPS waypoints stored as UTM coordinates5, the distance on-the-ground between two points might not be obvious from the coordinate sets. For example, what is the distance between (452632,4660214) and (452991,4660580), the marked locations of two observed dickcissel nests? Once we recognize these ordered pairs as the geographical equivalent of ( x , y ) pairs, it is pretty easy to see that the second nest is 452991 − 452632 = 359 m east and 4660580 − 4660214 = 366 m north of the first. But neither a dickcissel nor an ornithologist would likely go from one nest to another by first going 366 m north and then 359 m east. Both would be more likely to go in an approximately straight line. Since east and north are perpendicular, we can construct a triangle like Figure 9.4, with a 359 m long easting side and a 366 m long northing side to represent the coordinate distances. The as-the-crow-flies distance is the hypotenuse of the triangle since it is opposite the right angle. Therefore we can use the pythagorean theorem to find that distance, which we can call d :
\begin{array}{l}
d^{2}=(\text { easting })^{2}+(\text { northing })^{2} (9.2)\\
d=\sqrt{(\text { easting })^{2}+(\text { northing })^{2}} (9.3)\\
d=\sqrt{(359 m)^{2}+(366 m)^{2}}=513 \mathrm{~m} (9.4)
\end{array}
Universal Transverse Mercator, or UTM, refers to a projected geographic coordinate system wherein locations are given coordinates (meters easting, meters northing) according to their distance in meters east and north of a predefined datum. The benefit of UTM coordinates compared with latitude and longitude is that it is an orthogonal coordinate system analogous to the Cartesian x - y coordinate system we sometimes use for abstraction in math.
Knowing that the second nest is about 513 m away from the first is great. But if you were to give instructions to a field assistant to walk 513 m from the first nest to find the second nest, that alone is insufficient information to get to the correct place. Which direction does she have to go? You could, of course, simply have her walk north 366 m and then east 359 m, but that wouldn’t be terribly efficient. What is missing is obviously the direction. If she’s carrying a compass, you could give her a bearing or compass direction to follow, but what is that bearing, and do we have enough information to determine it?
9.2.3 Angles and azimuths
At this point, we need to draw more important distinctions regarding coordinate systems and conventions. When we need to be more accurate than simply saying “northeast”, compass bearings are of- ten given as angles in degrees. Some people prefer to use quadrant bearings, where directions are given with respect to deviations from north or south. For example, due northeast might be expressed as “north 45 east”, or equivalently N\(_{45}\)\(^{◦}\)E. That can be interpreted as 45\(^{◦}\) east of due north. Similarly, southeast could be S\(_{45}\)\(^{◦}\)E and southwest is S\(_{45}\)\(^{◦}\)W. This can sometimes be easier to understand in conversation, but bearings expressed in azimuth are less prone to misinterpretation. Azimuth is the compass direction in degrees clockwise from north, increasing continuously from 0 to 360 . In this system, north is both 0\(^{◦}\) and 360\(^{◦}\), east is 90\(^{◦}\), south is 180\(^{◦}\) and west is 270\(^{◦}\)(Figure 9.5).
In the world of mathematics, angles are usually measured counterclockwise from the x -axis . Not only does this mean that the starting point (0\(^{◦}\)) is in a different place, but it increases in a different direction. We will occasionally employ this convention, since it is so prevalent in quantitative topics unrelated to geography. But for the current problem, we’ll stick with azimuth.
Returning to the problem of finding the dickcissel nest, how can we decide what bearing to give to the field assistant? Since the easting and northing distances are similar, we can be fairly confident that it will be near 45\(^{◦}\) (NE), but probably not exactly that. But how do you determine the unknown angles in a triangle when you only know one angle (the right angle) and all of the side lengths? Aha! sine and cosine to the rescue!!
9.3 Angles, Circles and Sines
Before we proceed with finding the azimuth, let’s more formally define a few trigonometric quantities. We’ll do this initially in a mathematics framework, using an x - y coordinate system with angles increasing counterclockwise from the x -axis. Figure 9.6 shows what we might call the “unit circle”, a circle centered on the point (0,0) (also called the “origin”) with a radius of 1 unit. If we choose any point on the circle, call it p , it lies a distance of 1 unit from the origin. But its coordinates are not immediately obvious. As with the case of the dickcissel nests, however, we can break the path from the origin to p into a component in the x direction and a component in the y direction. Connecting each of those component paths with the direct path (the radius line), we end up with a right triangle △OAH, as illustrated in yellow in Figure 9.6.
Since we have a right triangle, we could use the Pythagorean theorem again to find an unknown side length, provided that we know two of the sides. But in this case, we do not know two sides. Instead we know the angle θ between the x -axis and the line connecting the origin with point p . We also know that this line, call it H for hypotenuse, has length 1 by definition. The other two sides, O and A for opposite and adjacent, are unknown but can be found from the fundamental trigonometric functions .
The
sine
of an angle is the ratio of the length of the opposite side (
O
) to the length of the hypotenuse side (
H
).
The
cosine
of an angle is the ratio of the length of the adjacent side (
A
) to the length of the hypotenuse side (
H
).
The
tangent
of an angle is the ratio of the length of the opposite side (
O
) to the length of the adjacent side (
A
).
The tangent is identical to the ratio of the sine to cosine of an angle, which you can see is equivalent to the above definition if you cancel the hypotenuse terms in the ratio of ratios. These are loaded definitions, so let’s take a few moments to ponder what we’ve just seen.
- The trig functions are functions in the formal sense: they convert an input (angle) to a unique output (ratio of side lengths).
- The opposite and adjacent sides are defined relative to the angle that is the argument of the trig function.
- The output of each trig function is a dimensionless quantity that represents the ratio of two side lengths.
- If we know one angle (other than the right angle) and one side length, we can find the two remaining side lengths using the trigonometric functions.
In equations, we don’t spell the full name of these functions, but instead use sin, cos, and tan as shorthand. With this shorthand and the definitions above, we can construct a few simple equations that can help us find the unknown side lengths O and A in Figure 9.6:
sin θ = \(\frac{O}{H}\) (9.5)
cos θ = \(\frac{A}{H}\) (9.6)
Since we know θ and H on the unit circle, we can rearrange these equations to solve for the unknowns. Multiplying both sides of each equation by H, we get:
H sin θ = O (9.7)
H cos θ = A (9.8)
Now the reason we have done this in the unit circle is that H = 1, so we essentially end with the definitions sin θ = O and cos θ = A . An important thing to realize, then, is that we can scale up to any arbitrary side lengths. Suppose we were interested in not a triangle with hypotenuse 1 unit, but one with hypotenuse 55 meters. Defining f = 55 and isometrically scaling all the triangle sides by that factor, we can see for example that:
\(\sin \theta=\frac{f O}{f H}\left[\frac{\mathrm{m}}{\mathrm{m}}\right]\) (9.9)
Of course the f ’s simply lengthen H and O by the same common factor, and could be easily canceled. But this illustrates the fact that sine and other trig functions describe dimensionless side-length ratios, and that those ratios can scale proportionally without changing the sine, cosine, and tangent of the angles ! An obvious implication is that if one vertex of a right triangle has the same cosine and sine as another triangle, the triangles can be shown to be similar.
Actually, we already knew that, since by knowing one angle other than the right angle in a right triangle we can easily find the third. Recall, then, that one of the criteria for identifying similarity in triangles is AAA, or equality of all three angles regardless of side length. For any pair of similar right triangles, the only difference in side lengths is a constant scaling factor f .
Thus, any right triangle can be scaled down to a similar one on the unit circle by dividing all side lengths by the length of the hypotenuse, so that H / f = 1!
In addition to simplifying triangle scaling, the unit circle also allows us to imagine moving point p along the perimeter and ob- serving how the lengths of A and O change accordingly. In fact, the software Geogebra is perfectly suited for doing this, and I highly recommend playing around with it to boost your intuition. The key thing to notice is that the denominator of the side ratios defined as the sine and cosine is the hypotenuse, or 1 on the unit circle. So the lengths of the opposite (sine) and adjacent (cosine) sides are the out- put of those respective functions. For angles between 0\(^{◦}\) and 90\(^{◦}\), both functions range from 0 to 1. If you allow x and y to take on neg- ative values as point p goes down or to the left of the origin, you’ll see that both functions remain between −1 and 1, inclusive.
But tangent is a different story. Recall that one definition for the tangent of an angle is
O
/
A
. You can see that for a small angle
θ
,
O
is quite small and
A
is pretty close to 1, so the ratio of the two will be nearly 0.
O
and
A
are equal and tan
θ
= 1 when
θ
is 45\(^{◦}\), and as
θ
approaches 90\(^{◦}\),
A
approaches 0 and
O
approaches infinity, so tan
θ
approaches infinity as well. What do you suppose happens as
θ
gets larger than 90\(^{◦}\)?
9.3.1 Example: tree clinometry
One of the most common ways to measure the height of a tree is with a clinometer. This is a small handheld device with a sighting lens and crosshair and one of a variety of different mechanisms for measuring the inclination angle (either positive or negative) of the sight-line from horizontal. Figure 9.7 illustrates the hypothetical triangles constructed with vertices at the observer’s eye, and the base and crown of the target tree. Obtaining the height distribution of merchantable timber in our forest parcel in Problem 3.5 could include a set of representative height measurements with a clinometer.
In measuring the height of a tree, two readings are often taken with the clinometer: one to the crown θ \(_{u}\) and one to the base or stump θ l . The observation point E is a pre-determined distance D from the tree itself. From this information, what is the height of the tree?
We identify
H
as the quantity of interest, and observe that
H
=
H
\(_{u}\)
+
H
\(_{l}\). As a first step, we must therefore determine
H
u
and
H
l
.
We assume the geometry of the problem allows us to construct two imaginary right triangles as illustrated and that our clinometer gives us angles in degrees from the horizontal. Since we know the horizontal distance to the tree
D
and have measured the angles to the top (
θ
\(_{u}\)) and bottom (
θ
\(_{l}\)) of the tree, we know the adjacent side length (
D
) and an angle for both triangles. The target unknowns are the opposite sides of each triangle, and from
sohcahtoa
we know that we can use tan to find the opposite sides when we know the adjacent sides.
Thus:
tan θ \(_{u}\) = \(\frac{H_u}{D}\) (9.10)
Dtan θ \(_{u}\) = H\(_{u}\) (9.11)
and
tan θ \(_{l}\) = \(\frac{H_l}{D}\) (9.12)
Dtan θ \(_{u}\) = H\(_{l}\) (9.13)
and since H = H \(_{u}\) + H \(_{l}\),
H = D tan θ \(_{l}\) + D tan θ \(_{l}\) (9.14)
H = D (tan θ \(_{u}\) + tan θ \(_{l}\)).
Thus, we can use an elementary trigonometric function (tan) and a bit of algebra to produce a formula that relates clinometer angle measurements to tree height.
9.4 Arbitrary triangles
While some problems may be approached profitably with imaginary right triangles, others present triangles without right angles. We’ll call these arbitrary or general triangles . Triangulation is a typical task in which ecologists or wildlife managers might encounter arbitrary triangles. When radio-tracking a collared animal, for example, one method for determining the animal’s location at a given time is by triangulation from multiple directional antennae. Figure 9.4 is the same as Figure 9.1, except that now we are considering the vertices to be radio transmitters (animals) and receivers (ecologists).
Thus far, we only have two tools that are safe to use with triangles that lack a right angle: the rule that all the interior angles sum to 180\(^{◦}\) (for which we may use the shorthand (Σ∠180) and the general criteria for and implications of similar triangles. These might be of limited use if our goal is to determine the distance to or location of a collared animal. But if we know the length of one side (say side b in Figure 9.4), and a few angles, we can make some progress.
9.4.1 Law of sines
The law of sines is valid for general triangles, including right triangles. If we are careful to define our sides and vertices as we have (with vertex angle A opposite side a and so on), we can state the law of sines as follows:
\(\frac{sin A}{a}\) = \(\frac{sin B}{b}\) = \(\frac{sin C}{c}\). (9.16)
Note that this equation is strange in that there are two equals signs. Don’t get worried, this is just a shorthand that allows us to say on one line that each ratio between the sine of an angle and the length of its opposite side is equal to the other corresponding sine/side ratios. If we wrote each equation with a single ‘=’, there would be three of them and it would simply take more space. But in actual implementation you can use any of the implied equalities in Equation 9.16, such as:
\(\frac{sin A}{a}\) = \(\frac{sin C}{c}\) (9.17)
The key concept here is that there is a simple and consistent relation- ship between the sine of each angle and the length of its opposite side, and that this applies to all triangles, no matter what size or shape . That makes plenty of sense right? If you imagine taking a triangle formed by three knotted rubber bands and lengthening one side (without changing the length of the other sides), what happens to the angle opposite that stretched side? It grows right? But to accommodate the growth of that angle, the other two angles must get smaller. The law of sines is especially helpful if you know two sides and one angle (SSA) or two angles and one side (AAS).
9.4.2 Law of cosines
Another tool that is useful for general triangles is called the “Law of cosines”. In many ways, it provides the same information that you can easily find from other tools we have already discussed, so we won’t derive it or discuss it in great detail. But mathematician Paul Lockhart makes the case that the law of cosines might be a misleading name, and that the relationship might be better described as a modified version of the Pythagorean theorem that is good for all general triangles. Check it out:
c \(^{2}\) = a \(^{2}\) + b \(^{2}\) −2 ab cos C . (9.18)
As you can see, it is identical to the Pythagorean theorem except that there is a correction factor 2 ab cos C that is subtracted to account for deviations from a right triangle. As with the Pythagorean theorem, the law of cosines gives you the third side length if you know the other two, but you also need to know the angle between the known sides (SSA). As such, it’s function overlaps that of the law of sines.
9.5 Triangle tools: a summary
Trigonometry is a large subdiscipline of mathematics, and can and does fill more than a semester in math classes. Our treatment here has focused on the tools that are most commonly encountered in practical field settings in the natural sciences. Many additional functions, relationships and skills can become important in specific, more technical applications, but most of these can be derived from the basic functions discussed here. These functions and properties are summarized in the table that follows.
9.5.1 Example: shoreline waterfowl habitat (Problem 3.1)
Some dabbling duck species like Mallards seem to prefer very shallow water. This means that small, shallow wetlands can fit the bill, but even the shallow shoreline areas of larger and deeper wetlands may be adequate. Shorelines are also the interface between feeding and nesting areas for many species, and often support diverse flora and fauna across the ecotone.
One way we could estimate the extent of shoreline habitat is to find the length of wetland perimeters, or outlines. If, as described in the last chapter, we have digitized (or obtained existing data for) the outlines of wetlands in our candidate parcels, we should have easting and northing coordinates for these outlines. As with polygon area, most GIS software will automatically compute the perimeter of any shape. However, it is instructive to see how this follows from our earlier discussion of triangle-assisted spatial reasoning.
Recall that when we were traversing between dickcissel nests, we used an implementation of the Pythagorean theorem to find the straight-line distance from the coordinates of both end points. We can write this relationship in an x , y coordinate system as follows:
\(l_{1 \rightarrow 2}=\sqrt{\left[\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}\right]}\) (9.19)
where l \(_{1 → 2}\) is the length of the straight-line distance from point 1 (with coordinates x \(_{1}\), y \(_{1}\)) and point 2. Notice that the result is always a real, non-negative number because the differences are squared. If we have a series of n points describing the wetland outlines, the sum of all n of the lengths forming a closed polygon approximate the perimeter P of the polygon\(^{7}\). We can generalize this as follows:
\(\begin{aligned}
P=\sqrt{\left[\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}\right]} &+\sqrt{\left[\left(x_{3}-x_{2}\right)^{2}+\left(y_{3}-y_{2}\right)^{2}\right]}+\ldots \\
& \ldots \sqrt{\left[\left(x_{1}-x_{n}\right)^{2}+\left(y_{1}-y_{n}\right)^{2}\right]} .
\end{aligned}\) (9.20)
As with the trapezoidal area formula, this can be implemented by hand, in a spreadsheet, or with GIS software.
\(^{7}\)How good is this sort of approximation of the perimeter? This is a simple question with a not-so-simple answer. For our purposes here, the more points we have, the better – particularly if we’ve got an automated way to do the computations. However, in a more philosophical sense this is the crux of the coastline paradox, first popularized by mathematician Benoit Mandelbrot.
1. Use Figure 9.4 and the right triangles formed by dropping the perpendicular h , to derive the law of sines from the trig functions you already know.
2. Explain how you could find the telemetered location of B if you know the locations of A and C and their internal angles.
3. What happens if you try to apply the law of sines to a right triangle?
4. What happens if you use the law of cosines on a right triangle? Assume angle C is the right angle.
5. Triangles can be used to measure distant objects, even if we can’t get to them. This can be used to estimate the height of an object (e.g., a mountain peak or tree) where we cannot access the base, and therefore cannot measure the full horizontal distance that separates us from the object we wish to measure. The general premise of the remote method is illustrated in figure 9.9. If we can use a clinometer (a device for determining the angle between horizontal and a sight-line to an object of interest) to determine the sight-line angles to the object of interest p from two different places b 1 and b 2, and we know the distance between those places l , we can use trigonometry and algebra to determine the desired height H .
Devise a strategy for measuring H from the information gathered at b 1 and b 2. Derive and justify a formula that can be used for this task.


10.12: Inductance (Exercise)
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Conceptual Questions
14.2 Mutual Inductance
1. Show that \(NΦ_m/I\) and \(ε/(dI/dt)\), which are both expressions for self-inductance, have the same units.
2. A 10-H inductor carries a current of 20 A. Describe how a 50-V emf can be induced across it.
3. The ignition circuit of an automobile is powered by a 12-V battery. How are we able to generate large voltages with this power source?
4. When the current through a large inductor is interrupted with a switch, an arc appears across the open terminals of the switch. Explain.
14.3 Self-Inductance and Inductors
5. Does self-inductance depend on the value of the magnetic flux? Does it depend on the current through the wire? Correlate your answers with the equation \(NΦ_m=LI\).
6. Would the self-inductance of a 1.0 m long, tightly wound solenoid differ from the self-inductance per meter of an infinite, but otherwise identical, solenoid?
7. Discuss how you might determine the self-inductance per unit length of a long, straight wire.
8. The self-inductance of a coil is zero if there is no current passing through the windings. True or false?
9. How does the self-inductance per unit length near the center of a solenoid (away from the ends) compare with its value near the end of the solenoid?
14.5 RL Circuits
11. Use Lenz’s law to explain why the initial current in the RL circuit of Figure 14.12(b) is zero.
12. When the current in the RL circuit of Figure 14.12(b) reaches its final value \(ε/R\), what is the voltage across the inductor? Across the resistor?
13. Does the time required for the current in an RL circuit to reach any fraction of its steady-state value depend on the emf of the battery?
14. An inductor is connected across the terminals of a battery. Does the current that eventually flows through the inductor depend on the internal resistance of the battery? Does the time required for the current to reach its final value depend on this resistance?
15. At what time is the voltage across the inductor of the RL circuit of Figure 14.12(b) a maximum?
16. In the simple RL circuit of Figure 14.12(b), can the emf induced across the inductor ever be greater than the emf of the battery used to produce the current?
17. If the emf of the battery of Figure 14.12(b) is reduced by a factor of 2, by how much does the steady-state energy stored in the magnetic field of the inductor change?
18. A steady current flows through a circuit with a large inductive time constant. When a switch in the circuit is opened, a large spark occurs across the terminals of the switch. Explain.
19. Describe how the currents through \(R_1\) and \(R_2\) shown below vary with time after switch S is closed.
20. Discuss possible practical applications of RL circuits.
14.6 Oscillations in an LC Circuit
21. Do Kirchhoff’s rules apply to circuits that contain inductors and capacitors?
22. Can a circuit element have both capacitance and inductance?
23. In an LC circuit, what determines the frequency and the amplitude of the energy oscillations in either the inductor or capacitor?
14.7 RLC Series Circuits
24. When a wire is connected between the two ends of a solenoid, the resulting circuit can oscillate like an RLC circuit. Describe what causes the capacitance in this circuit.
25. Describe what effect the resistance of the connecting wires has on an oscillating LC circuit.
26. Suppose you wanted to design an LC circuit with a frequency of 0.01 Hz. What problems might you encounter?
27. A radio receiver uses an RLC circuit to pick out particular frequencies to listen to in your house or car without hearing other unwanted frequencies. How would someone design such a circuit?
Problems
14.2 Mutual Inductance
28. When the current in one coil changes at a rate of 5.6 A/s, an emf of \(6.3×10^{−3}V\). is induced in a second, nearby coil. What is the mutual inductance of the two coils?
29. An emf of \(9.7×10^{−3}V\) is induced in a coil while the current in a nearby coil is decreasing at a rate of 2.7 A/s. What is the mutual inductance of the two coils?
30. Two coils close to each other have a mutual inductance of 32 mH. If the current in one coil decays according to \(I=I_0e^{−αt}\), where \(I_0=5.0A\) and \(α=2.0×10^3s^{−1}\), what is the emf induced in the second coil immediately after the current starts to decay? At \(t=1.0×10^{−3}s\)?
31. A coil of 40 turns is wrapped around a long solenoid of cross-sectional area \(7.5×10^{−3}m^2\). The solenoid is 0.50 m long and has 500 turns.
(a) What is the mutual inductance of this system?
(b) The outer coil is replaced by a coil of 40 turns whose radius is three times that of the solenoid. What is the mutual inductance of this configuration?
32. A 600-turn solenoid is 0.55 m long and 4.2 cm in diameter. Inside the solenoid, a small (1.1cm×1.4cm), single-turn rectangular coil is fixed in place with its face perpendicular to the long axis of the solenoid. What is the mutual inductance of this system?
33. A toroidal coil has a mean radius of 16 cm and a cross-sectional area of \(0.25cm^2\); it is wound uniformly with 1000 turns. A second toroidal coil of 750 turns is wound uniformly over the first coil. Ignoring the variation of the magnetic field within a toroid, determine the mutual inductance of the two coils.
34. A solenoid of \(N_1\) turns has length \(l_1\) and radius \(R_1\), and a second smaller solenoid of \(N_2\) turns has length \(l_2\) and radius \(R_2\). The smaller solenoid is placed completely inside the larger solenoid so that their long axes coincide. What is the mutual inductance of the two solenoids?
14.3 Self-Inductance and Inductors
35. An emf of 0.40 V is induced across a coil when the current through it changes uniformly from 0.10 to 0.60 A in 0.30 s. What is the self-inductance of the coil?
36. The current shown in part (a) below is increasing, whereas that shown in part (b) is decreasing. In each case, determine which end of the inductor is at the higher potential.
37. What is the rate at which the current though a 0.30-H coil is changing if an emf of 0.12 V is induced across the coil?
38. When a camera uses a flash, a fully charged capacitor discharges through an inductor. In what time must the 0.100-A current through a 2.00-mH inductor be switched on or off to induce a 500-V emf?
39. A coil with a self-inductance of 2.0 H carries a current that varies with time according to \(I(t)=(2.0A)sin120πt\). Find an expression for the emf induced in the coil.
40. A solenoid 50 cm long is wound with 500 turns of wire. The cross-sectional area of the coil is \(2.0cm^2\) What is the self-inductance of the solenoid?
41. A coil with a self-inductance of 3.0 H carries a current that decreases at a uniform rate \(dI/dt=−0.050A/s\). What is the emf induced in the coil? Describe the polarity of the induced emf.
42. The current \(I(t)\) through a 5.0-mH inductor varies with time, as shown below. The resistance of the inductor is 5.0Ω . Calculate the voltage across the inductor at \(t=2.0ms,t=4.0ms,\) and \(t=8.0ms\).
43. A long, cylindrical solenoid with 100 turns per centimeter has a radius of 1.5 cm. (a) Neglecting end effects, what is the self-inductance per unit length of the solenoid? (b) If the current through the solenoid changes at the rate 5.0 A/s, what is the emf induced per unit length?
44. Suppose that a rectangular toroid has 2000 windings and a self-inductance of 0.040 H. If h=0.10mh=0.10m, what is the ratio of its outer radius to its inner radius?
45. What is the self-inductance per meter of a coaxial cable whose inner radius is 0.50 mm and whose outer radius is 4.00 mm?
14.4 Energy in a Magnetic Field
46. At the instant a current of 0.20 A is flowing through a coil of wire, the energy stored in its magnetic field is \(6.0×10^{−3}J\). What is the self-inductance of the coil?
47. Suppose that a rectangular toroid has 2000 windings and a self-inductance of 0.040 H. If \(h=0.10m\), what is the current flowing through a rectangular toroid when the energy in its magnetic field is \(2.0×10^{−6}J\)?
48. Solenoid A is tightly wound while solenoid B has windings that are evenly spaced with a gap equal to the diameter of the wire. The solenoids are otherwise identical. Determine the ratio of the energies stored per unit length of these solenoids when the same current flows through each.
49. A 10-H inductor carries a current of 20 A. How much ice at 0°C could be melted by the energy stored in the magnetic field of the inductor? (Hint: Use the value \(L_f=334J/g\) for ice.)
50. A coil with a self-inductance of 3.0 H and a resistance of 100Ω carries a steady current of 2.0 A. (a) What is the energy stored in the magnetic field of the coil? (b) What is the energy per second dissipated in the resistance of the coil?
51. A current of 1.2 A is flowing in a coaxial cable whose outer radius is five times its inner radius. What is the magnetic field energy stored in a 3.0-m length of the cable?
14.5 RL Circuits
52. In Figure 14.12, \(ε=12V, L=20mH,\) and \(R=5.0Ω\). Determine (a) the time constant of the circuit, (b) the initial current through the resistor, (c) the final current through the resistor, (d) the current through the resistor when \(t=2τ_L\), and (e) the voltages across the inductor and the resistor when \(t=2τ_L\).
53. For the circuit shown below, \(ε=20V, L=4.0mH,\) and \(R=5.0Ω\). After steady state is reached with \(S_1\) closed and \(S_2\) open, \(S_2\) is closed and immediately thereafter (at \(t=0\)) \(S_1\) is opened. Determine (a) the current through \(L\) at \(t=0\), (b) the current through \(L\) at \(t=4.0×10^{−4}s\), and (c) the voltages across \(L\) and \(R\) at \(t=4.0×10^{−4}s\). \(R_1 = R_2 = R\).
54. The current in the RL circuit shown here increases to 40% of its steady-state value in 2.0 s. What is the time constant of the circuit?
55. How long after switch \(S_1\) is thrown does it take the current in the circuit shown to reach half its maximum value? Express your answer in terms of the time constant of the circuit.
56. Examine the circuit shown below in part (a). Determine \(dI/dt\) at the instant after the switch is thrown in the circuit of
(a), thereby producing the circuit of
(b). Show that if I were to continue to increase at this initial rate, it would reach its maximum \(ε/R\) in one time constant.
57. The current in the RL circuit shown below reaches half its maximum value in 1.75 ms after the switch \(S_1\) is thrown. Determine
(a) the time constant of the circuit and
(b) the resistance of the circuit if \(L=250mH\).
58. Consider the circuit shown below. Find \(I_1,I_2,\) and \(I_3\) when
(a) the switch S is first closed,
(b) after the currents have reached steady-state values, and
(c) at the instant the switch is reopened (after being closed for a long time).
59. For the circuit shown below, \(ε=50V, R_1=10Ω, R_2=R_3=19.4Ω\) and \(L=2.0mH\). Find the values of \(I_1\) and \(I_2\)
(a) immediately after switch S is closed,
(b) a long time after S is closed,
(c) immediately after S is reopened, and
(d) a long time after S is reopened.
60. For the circuit shown below, find the current through the inductor \(2.0×10^{−5}s\) after the switch is reopened.
61. Show that for the circuit shown below, the initial energy stored in the inductor, \(LI^2(0)/2\), is equal to the total energy eventually dissipated in the resistor, \(∫^∞_0I^2(t)Rdt\).
14.6 Oscillations in an LC Circuit
62. A 5000-pF capacitor is charged to 100 V and then quickly connected to an 80-mH inductor. Determine
(a) the maximum energy stored in the magnetic field of the inductor,
(b) the peak value of the current, and
(c) the frequency of oscillation of the circuit.
63. The self-inductance and capacitance of an LC circuit are 0.20 mH and 5.0 pF. What is the angular frequency at which the circuit oscillates?
64. What is the self-inductance of an LC circuit that oscillates at 60 Hz when the capacitance is \(10μF\)?
65. In an oscillating LC circuit, the maximum charge on the capacitor is \(2.0×10^{−6}C\) and the maximum current through the inductor is 8.0 mA.
(a) What is the period of the oscillations?
(b) How much time elapses between an instant when the capacitor is uncharged and the next instant when it is fully charged?
66. The self-inductance and capacitance of an oscillating LC circuit are \(L=20mH\) and \(C=1.0μF\), respectively.
(a) What is the frequency of the oscillations?
(b) If the maximum potential difference between the plates of the capacitor is 50 V, what is the maximum current in the circuit?
67. In an oscillating LC circuit, the maximum charge on the capacitor is \(q_m\). Determine the charge on the capacitor and the current through the inductor when energy is shared equally between the electric and magnetic fields. Express your answer in terms of \(q_m, L,\) and \(C\).
68. In the circuit shown below, \(S_1\) is opened and \(S_2\) is closed simultaneously. Determine
(a) the frequency of the resulting oscillations,
(b) the maximum charge on the capacitor,
(c) the maximum current through the inductor, and
(d) the electromagnetic energy of the oscillating circuit.
69. An LC circuit in an AM tuner (in a car stereo) uses a coil with an inductance of 2.5 mH and a variable capacitor. If the natural frequency of the circuit is to be adjustable over the range 540 to 1600 kHz (the AM broadcast band), what range of capacitance is required?
14.7 RLC Series Circuits
70. In an oscillating RLC circuit, \(R=5.0Ω,L=5.0mH\), and \(C=500μF\). What is the angular frequency of the oscillations?
71. In an oscillating RLC circuit with \(L=10mH,C=1.5µF,\) and \(R=2.0Ω\), how much time elapses before the amplitude of the oscillations drops to half its initial value?
72. What resistance R must be connected in series with a 200-mH inductor of the resulting RLC oscillating circuit is to decay to 50% of its initial value of charge in 50 cycles? To 0.10% of its initial value in 50 cycles?
Additional Problems
73. Show that the self-inductance per unit length of an infinite, straight, thin wire is infinite.
74. Two long, parallel wires carry equal currents in opposite directions. The radius of each wire is a, and the distance between the centers of the wires is d. Show that if the magnetic flux within the wires themselves can be ignored, the self-inductance of a length l of such a pair of wires is \(L=\frac{μ_0l}{π}ln\frac{d−a}{a}\). (Hint: Calculate the magnetic flux through a rectangle of length l between the wires and then use \(L=NΦ/I\).)
75. A small, rectangular single loop of wire with dimensions l, and a is placed, as shown below, in the plane of a much larger, rectangular single loop of wire. The two short sides of the larger loop are so far from the smaller loop that their magnetic fields over the smaller fields over the smaller loop can be ignored. What is the mutual inductance of the two loops?
76. Suppose that a cylindrical solenoid is wrapped around a core of iron whose magnetic susceptibility is \(x\). Using Equation 14.9, show that the self-inductance of the solenoid is given by \(L=\frac{(1+x)μ_0N^2A}{l}\), where I is its length, A its cross-sectional area, and N its total number of turns.
77. The solenoid of the preceding problem is wrapped around an iron core whose magnetic susceptibility is \(4.0×10^3\).
(a) If a current of 2.0 A flows through the solenoid, what is the magnetic field in the iron core?
(b) What is the effective surface current formed by the aligned atomic current loops in the iron core?
(c) What is the self-inductance of the filled solenoid?
78. A rectangular toroid with inner radius \(R_1=7.0cm\), outer radius \(R_2=9.0cm\), height \(h=3.0\), and \(N=3000\) turns is filled with an iron core of magnetic susceptibility \(5.2×10^3\).
(a) What is the self-inductance of the toroid?
(b) If the current through the toroid is 2.0 A, what is the magnetic field at the center of the core?
(c) For this same 2.0-A current, what is the effective surface current formed by the aligned atomic current loops in the iron core?
79. The switch S of the circuit shown below is closed at \(t=0\). Determine (a) the initial current through the battery and (b) the steady-state current through the battery.
80. In an oscillating RLC circuit, \(R=7.0Ω,L=10mH,\) and \(C=3.0μF\). Initially, the capacitor has a charge of \(8.0μC\) and the current is zero. Calculate the charge on the capacitor
(a) five cycles later and
(b) 50 cycles later.
81. A 25.0-H inductor has 100 A of current turned off in 1.00 ms.
(a) What voltage is induced to oppose this?
(b) What is unreasonable about this result?
(c) Which assumption or premise is responsible?
Challenge Problems
82. A coaxial cable has an inner conductor of radius a, and outer thin cylindrical shell of radius b. A current I flows in the inner conductor and returns in the outer conductor. The self-inductance of the structure will depend on how the current in the inner cylinder tends to be distributed. Investigate the following two extreme cases.
(a) Let current in the inner conductor be distributed only on the surface and find the self-inductance.
(b) Let current in the inner cylinder be distributed uniformly over its cross-section and find the self-inductance. Compare with your results in (a).
83. In a damped oscillating circuit the energy is dissipated in the resistor. The Q -factor is a measure of the persistence of the oscillator against the dissipative loss. (a) Prove that for a lightly damped circuit the energy, U , in the circuit decreases according to the following equation. \(\frac{dU}{dt}=−2βU\), where \(β=\frac{R}{2L}\). (b) Using the definition of the Q -factor as energy divided by the loss over the next cycle, prove that Q -factor of a lightly damped oscillator as defined in this problem is \(Q≡\frac{U_{begin}}{ΔU_{one \: cycle}} = \frac{1}{2 \pi R}\sqrt{\frac{L}{C}}\). (Hint: For (b), to obtain Q , divide E at the beginning of one cycle by the change \(ΔE\) over the next cycle.)
84. The switch in the circuit shown below is closed at \(t=0s\). Find currents through (a) \(R_1\), (b) \(R_2\), and (c) the battery as function of time.
85. A square loop of side 2 cm is placed 1 cm from a long wire carrying a current that varies with time at a constant rate of 3 A/s as shown below.
(a) Use Ampère’s law and find the magnetic field as a function of time from the current in the wire.
(b) Determine the magnetic flux through the loop.
(c) If the loop has a resistance of 3Ω , how much induced current flows in the loop?
86. A rectangular copper ring, of mass 100 g and resistance 0.2Ω , is in a region of uniform magnetic field that is perpendicular to the area enclosed by the ring and horizontal to Earth’s surface. The ring is let go from rest when it is at the edge of the nonzero magnetic field region (see below).
(a) Find its speed when the ring just exits the region of uniform magnetic field.
(b) If it was let go at t=0t=0, what is the time when it exits the region of magnetic field for the following values: \(a=25cm, b=50cm,B=3T,\) and \(g=9.8m/s^2\)? Assume the magnetic field of the induced current is negligible compared to 3 T.


8.2: Cell Division
By the end of this section, you will be able to do the following:
- Describe the structure of prokaryotic and eukaryotic genomes
- Distinguish between chromosomes, genes, and traits
- Describe the mechanisms of chromosome compaction
The continuity of life from one cell to another has its foundation in the reproduction of cells by way of the cell cycle. The cell cycle is an orderly sequence of events that describes the stages of a cell’s life from the division of a single parent cell to the production of two new genetically identical daughter cells.
Genomic DNA
Before discussing the steps a cell must undertake to replicate and divide its DNA, a deeper understanding of the structure and function of a cell’s genetic information is necessary. A cell’s DNA, packaged as a double-stranded DNA molecule, is called its genome . In prokaryotes, the genome is composed of a single, double-stranded DNA molecule in the form of a loop or circle (Figure \(\PageIndex{1}\)). The region in the cell containing this genetic material is called a nucleoid. Some prokaryotes also have smaller loops of DNA called plasmids that are not essential for normal growth. Bacteria can exchange these plasmids with other bacteria, sometimes receiving beneficial new genes that the recipient can add to their chromosomal DNA. Antibiotic resistance is one trait that often spreads through a bacterial colony through plasmid exchange from resistant donors to recipient cells.
In eukaryotes, the genome consists of several double-stranded linear DNA molecules (Figure \(\PageIndex{2}\)). Each species of eukaryotes has a characteristic number of chromosomes in the nuclei of its cells. Human body (somatic) cells have 46 chromosomes, while human gametes (sperm or eggs) have 23 chromosomes each. A typical body cell contains two matched or homologous sets of chromosomes (one set from each biological parent)—a configuration known as diploid . (Note: The letter n is used to represent a single set of chromosomes; therefore, a diploid organism is designated 2 n .) Human cells that contain one set of chromosomes are called gametes, or sex cells; these are eggs and sperm, and are designated 1n , or haploid .
Upon fertilization, each gamete contributes one set of chromosomes, creating a diploid cell containing matched pairs of chromosomes called homologous (“same knowledge”) chromosomes . Homologous chromosomes are the same length and have specific nucleotide segments called genes in exactly the same location, or locus . Genes, the functional units of chromosomes, determine specific characteristics by coding for specific proteins. Traits are the variations of those characteristics. For example, hair color is a characteristic with traits that are blonde, brown, or black, and many colors in between.
Each copy of a homologous pair of chromosomes originates from a different parent; therefore, the different genes (alleles) themselves are not identical, although they code for the same traits such as “hair color.” The variation of individuals within a species is due to the specific combination of the genes inherited from both parents. Even a slightly altered sequence of nucleotides within a gene can result in an alternative trait. For example, there are three possible gene sequences on the human chromosome that code for blood type: sequence A, sequence B, and sequence O. Because all diploid human cells have two copies of the chromosome that determines blood type, the blood type (the trait) is determined by the two alleles of the marker gene that are inherited. It is possible to have two copies of the same gene sequence on both homologous chromosomes, with one on each (for example, AA, BB, or OO), or two different sequences, such as AB, AO, or BO.
Apparently minor variations of traits, such as blood type, eye color, and handedness, contribute to the natural variation found within a species, but even though they seem minor, these traits may be connected with the expression of other traits as of yet unknown. However, if the entire DNA sequence from any pair of human homologous chromosomes is compared, the difference is much less than one percent.
Eukaryotic Chromosomal Structure and Compaction
If the DNA from all 46 chromosomes in a human cell nucleus were laid out end-to-end, it would measure approximately two meters; however, its diameter would be only 2 nm! Considering that the size of a typical human cell is about 10 µm (100,000 cells lined up to equal one meter), DNA must be tightly packaged to fit in the cell’s nucleus. At the same time, it must also be readily accessible for the genes to be expressed. For this reason, the long strands of DNA are condensed into compact chromosomes during certain stages of the cell cycle. There are a number of ways that chromosomes are compacted.
In the first level of compaction, short stretches of the DNA double helix wrap around a core of eight histone proteins at regular intervals along the entire length of the chromosome (Figure \(\PageIndex{3}\)). The DNA-histone complex is called chromatin. The beadlike, histone DNA complex is called a nucleosome , and DNA connecting the nucleosomes is called linker DNA. A DNA molecule in this form is about seven times shorter than the double helix without the histones, and the beads are about 10 nm in diameter, in contrast with the 2-nm diameter of a DNA double helix.
The second level of compaction occurs as the nucleosomes and the linker DNA between them coil into a 30-nm chromatin fiber. This coiling further condenses the chromosome so that it is now about 50 times shorter than the extended form.
In the third level of compaction, a variety of fibrous proteins is used to “pack the chromatin.” These fibrous proteins also ensure that each chromosome in a non-dividing cell occupies a particular area of the nucleus that does not overlap with that of any other chromosome (see the top image in Figure \(\PageIndex{3}\)).
DNA replicates in the S phase of interphase, which technically is not a part of mitosis, but must always precede it . After replication, the chromosomes are composed of two linked sister chromatids . When fully compact, the pairs of identically packed chromosomes are bound to each other by cohesin proteins. The connection between the sister chromatids is closest in a region called the centromere . The conjoined sister chromatids, with a diameter of about 1 µm, are visible under a light microscope. The centromeric region is highly condensed and thus will appear as a constricted area.
Link to Learning
Link to Learning
This animation illustrates the different levels of chromosome packing.
Contributors and Attributions
Remixed and/or curated from the following works:
Clark, M. A., Douglas, M., & Choi, J. (2018). 2.7.2 Cell Division . In Biology 2e . OpenStax (CC BY 4.0; Access for free at https://openstax.org/books/biology-2e/pages/1-introduction ).


2.10: NAICS Codes and Direct Competitors
- Use NAICS codes to determine direct competitors
The North American Industry Classification System (NAICS), which replaced the Standard Industrial Classification (SIC) system in 1997, is used by federal statistical agencies to classify businesses. They do this so that data collected on the businesses can be aggregated to create a snapshot of the total industry for analysis and comparison. Some common NAICS codes for food retailers are as follows:
-
445110—supermarkets and other grocery (except convenience) stores
- commissaries, primarily grocery
- delicatessens, primarily retailing a wide variety of grocery items and meat
- food stores
- grocery stores
- supermarkets
- 445120—convenience stores
- 445210—meat markets
- 445220—fish and seafood markets
- 445230—fruit and vegetable markets
- 446110—pharmacies and drug stores
- 452311—warehouse clubs and supercenters
- 452319—all other general merchandise stores
Please note, however, that the above list is not exhaustive. For example, there are unique codes for candy stores, stores that sell nuts, bakeries that bake goods on premises, and bakeries that don’t. What’s important is that you can identify your NAICS code to see statistics about your specific industry, allowing you to get a better understanding of your competitive set.
Contributors and Attributions
- NAICS Codes and Direct Competitors. Authored by : Patrick Williams. Provided by : Lumen Learning. License : CC BY: Attribution


1.4: EKG Machinery and Setup
Note: There are many, many types of EKG electrode patches. Some are a simple sticker where the adhesive and gel pad are combined, some have a small metal button or clip where the wire for the leads attach. This EKG learning module does not endorse any specific brand or type of EKG electrode or patch.
The gel pad on the above EKG electrode improves conduction from the surface of the chest to the lead. Most electrodes have a similar compound and mechanism to improve the lead’s ability to detect electrical activity.
The electrode also incorporates an adhesive that sticks the pad to the chest wall. The adhesive alone is usually adequate to adhere the electrode well, but a poorly adhered electrode will not detect electrical activity adequately (or at all). Common confounding factors that can make it harder for the electrode to adhere to the chest wall include excessive lotion or topical oil used on the skin, extensive sweating or diaphoresis, and hair. It is very acceptable and encouraged to shave or clip the chest of a patient who needs an EKG but has too much hair to allow the electrodes to stick. It is also acceptable to use tape or other adhesives to reinforce the electrode’s adhesive.
There are many, many types of monitors that serve a variety of purposes. Some units are designed for monitoring purposes only, others are multi-functional and can include many features like defibrillation, external pacing, and monitoring of other clinical data like blood pressure and end tidal CO2. This EKG interpretation lesson series does not endorse any one EKG machine type or brand over the others.
While the design of the machines are very different, their purposes as EKG monitors are all functionally the same: they all measure and interpret the flow of electrical activity. This course does not endorse or recommend any specific type of EKG monitor or machine over others.
Most hospitals monitor patients via at least 3 leads simultaneously. An EKG lead is a particular electrical “view” of the heart and is representative of flow of electricity toward and away from 2 points in the heart. The setup looks like this:
Each lead is named and will be labeled: RA, LA, RL, and LL. The easiest way to remember this is RA = right arm, LA = left arm, RL = right leg, LL = left leg. The leads can be placed on the extremities or as is pictured on the corresponding corner of the trunk.
Lead II is the most commonly interpreted with regards to rhythm and is usually the first lead students learn to interpret. It goes diagonally from the right upper chest to the left lower chest in the same positions a defibrillator or AED would be in. Lead I goes laterally across the chest, measuring electricity from a positive electrode on the left upper chest and a negative electrode on the right upper chest. Lead III measures vertically across the left chest wall. Its negative electrode is the left upper chest and its positive electrode is the left lower chest. Notice that the positive electrode for lead I is the same as the negative for lead III - every physical electrode placed is capable of measuring both positively and negatively in the way the left upper chest electrode does.
The above picture is normal sinus rhythm in leads I, II, and III. Notice the differences for each. The P waves in leads I and II are upright, as electricity in the heart flows primarily toward their positive leads. The P wave in lead III however is biphasic and somewhat flattened. That is because electricity flows toward both the positive and negative electrodes (as the electricity flows in a down-left manner, and both positive and negative electrodes are to the left of the heart). The QRS complexes have similar morphological differences due to the differences in lead positioning.
Einthoven’s Triangle
A standard 12 lead EKG measures 12 “views” of the heart using 10 electrodes. 6 of the leads are bipolar, which means they have both an electrode measuring both positive and negative activity. Thanks to mathematical calculations, these 6 bipolar leads only require 4 electrodes: the 3 pictured above and a ground electrode, which helps remove interference and artifact (the above setup is possible without the ground electrode). 3 of the leads are considered to be “virtual leads,” or leads that are not physically on the patient but have been calculated mathematically based on the 3 existing electrodes. The triangular arrangement of electrodes is referred to as Einthoven’s triangle.
The virtual leads are aVL, aVR, and aVF. You can remember which is which by remembering that VL is the virtual left lead, VR is the virtual right lead, and VF is the virtual in F erior lead. It can be difficult to visualize, but the positive electrode for aVR is RA, the negative electrode is a virtual electrode that is directly between LA and LL. This lead is calculated using a formula based on the electrical activity tracked on the LA and LL electrodes. Similarly, aVL uses LA as a positive electrode and uses a virtual electrode between RA and LL that is mathematically calculated as a negative electrode. The aVF lead uses LL as a positive electrode and a virtual negative electrode between RA and LA.
Above is a normal sinus rhythm in leads aVR, aVL, and aVF. Notice that aVR is almost a perfect inversion of lead II. Refer to Einthoven’s triangle:
Notice that lead II and aVR are almost (but not completely) parallel in their orientation, but their direction is opposite? The negative lead for lead II is the positive lead for aVR, and the virtual lead between LL and LA is the negative lead for aVR. Lead LL is the positive lead for lead II. Compare all 6 of the bipolar leads in this normal sinus rhythm EKG:
II and aVR are almost perfect inversions, which makes sense when you consider the lead orientation as we did above. II, III and aVF are also very similar, which also makes sense because the LL lead is the positive electrode for all 3 of these leads. The negative virtual electrode for aVF is directly between the negative electrodes for II (RA) and III (LA). Notice that the morphology of the entire aVF lead is essentially an intermediary of II and III: the P wave for aVF is taller than it is in lead III but shorter than it is in lead II. The QRS is complex has an amplitude (height) that is taller than III but shorter than II, the S in particular is not as low in amplitude (deep) as in lead II but is lower than in lead III. The T wave is also larger than in lead III, but smaller than lead II.
aVL is a stranger lead if directly compared to I, II, and III. While it does have some things in common with lead I, including a shared positive electrode (LA), its QRS complex is more biphasic (goes both positively and negatively) than lead I. The P wave is also flatter, and the T wave is barely visible. This has to do with the placement of the negative virtual electrode, between the RA and LL electrodes; the electricity flows simultaneously in both directions, and the amount of electricity going toward both electrodes is closer to equal than it is in lead I. This leads to flattened P and T waves because simultaneous positive and negative electrical flow registers on the EKG graph as flat. This is because the computer balances the negative against the positive, so for example electricity measuring 3 mV going toward the positive electrode would normally register as an upward inflection on the EKG that is 3 small boxes high. However, if there is also 3 mV of electricity going toward the negative electrode at the same time, the computer interprets this as 0 neg amplitude because the 3 mV flowing towards the negative electrode cancels out the 3 mV flowing towards the positive.
EKG interpretation in multiple leads can be more challenging because there is more to consider regarding electrical flow, directionality, and how the leads interpret the electrical flow. It is important to be able to recognize at least basic features of the EKG in each lead, as well as what is normal for each lead, because most arrhythmias need to be confirmed in multiple leads. Confirmation in more than one lead is important because EKG machines, electrodes, the wires themselves, and placement of electrodes can all cause abnormal readings, and confirming an arrhythmia in another lead is done to ensure an arrhythmia is, in fact, an electrical abnormality with the heart and not a mechanical malfunction or human error in lead placement.
Lead Reversals
Human error is common in obtaining EKGs, and while lead placement is not always a precise measurement, it is important to double check the correct wires are connected to the correct electrodes. If you accidentally mixed 2 of your leads up, there are also morphological changes to the EKG that are predictable, identifiable as a lead reversal, and can be cause for alarm if they are not correctly identified as a lead reversal.
If you reverse the LA and RA leads, for example, lead I inverts:
This is because the negative RA electrode is now in the LA position, and the positive LA electrode is now in the RA position in error. You will also notice that leads II and III have switched: lead III’s negative electrode, LA, is now in the RA position (normally lead II’s negative electrode), and lead II’s negative electrode RA is now in the LA position. The differences in leads II and III are not particularly marked in this example of lead reversal, but you will notice that the P and T waves in lead III are taller than they are in lead II which is normally opposite. The virtual leads aVL and aVR also reverse (not pictured in EKG graph), so aVL will look like aVR normally does and vice versa.
If you reverse the LA and LL leads, this is the result:
Lead III is inverted, which makes sense because LA is normally the negative electrode and LL is normally the positive electrode for this lead. When they are reversed, you will notice all of the waves are inverted compared to correct placement. You will also notice that lead I looks more like lead II with a taller P and T wave; that’s because the normal leads for lead I are now in the position they should be for lead II (RA in the correct position, LA in the position of LL). Lead II is essentially now lead I, as its positive lead (LL) is incorrectly in the LA position.
All of the leads, when reversed, will show predictable changes that have to do with the physical lead placement. This primer will not go into deeper detail about lead reversal, but know that there will be at least subtle changes in most of the leads if there are lead reversals, as most modern EKG machines interpret data and use mathematical calculations based on the physical leads to predict virtual leads, like in aVR, aVL, and aVF. It is also important to consider that correct electrode placement, with or without lead reversal issues, is important for the same reason.
The first 4 leads, including the ground lead (RL) are the same as they are for our previous readings. This time, we are adding 6 additional unipolar leads, labeled V1 through V6 on the diagram. Unipolar leads are a little different from bipolar leads. A bipolar lead measures electricity in 2 directions - toward a positive electrode and toward a negative electrode. A unipolar lead only measures toward a positive lead. You will notice that, despite not having a negative lead, leads V1 through V6 do have negative deflections on them in the same way leads I through aVF do. This is because the “negative electrode” in this case is similar to the virtual electrodes in leads aVR-aVF. The computer uses a mathematical calculation of all of the other leads to produce a virtual electrode. This time, the virtual electrode is intended to be near the center of the heart. If you compare it to an anatomic cross-section, the V leads evaluate the transverse plane while leads I-aVF evaluate the coronal plane:
It is important to make sure leads are placed correctly for a 12 lead. V1 is the only lead that is located to the right of the sternum. V1 and V2 are placed at the 4th intercostal space to the right and left of the sternum, respectively. It is recommended to physically palpate the first 4 intercostal spaces when you place leads V1 and V2; the location is anatomically lower on the chest wall than it appears at a glance. V3 and V4 are placed next, both in the 4th intercostal space. V4 should be placed in the 4th intercostal space directly below the nipple - this is also about the mid-clavicular line. V3 is placed in the 4th intercostal space directly, but laterally placed halfway between V2 and V4.
V5 and V6 are placed in the 5th intercostal space. V6 is placed horizontally along the mid-axillary line in the 5th intercostal space. V5 is placed in the 5th intercostal space, halfway between V4 and V6.


14: Genes and Proteins
Since the rediscovery of Mendel’s work in 1900, the definition of the gene has progressed from an abstract unit of heredity to a tangible molecular entity capable of replication, expression, and mutation. Genes are composed of DNA and are linearly arranged on chromosomes. Genes specify the sequences of amino acids, which are the building blocks of proteins. In turn, proteins are responsible for orchestrating nearly every function of the cell. Both genes and the proteins they encode are absolutely essential to life as we know it.
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- 14.1: Introduction to Genes and Proteins
- Since the rediscovery of Mendel’s work in 1900, the definition of the gene has progressed from an abstract unit of heredity to a tangible molecular entity capable of replication, expression, and mutation. Genes are composed of DNA and are linearly arranged on chromosomes. Genes specify the sequences of amino acids, which are the building blocks of proteins. In turn, proteins are responsible for orchestrating nearly every function of the cell. Both genes and the proteins they encode are absolutel
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- 14.2: The Genetic Code
- The cellular process of transcription generates messenger RNA (mRNA), a mobile molecular copy of one or more genes with an alphabet of A, C, G, and uracil (U). Translation of the mRNA template converts nucleotide-based genetic information into a protein product. Protein sequences consist of 20 commonly occurring amino acids; therefore, it can be said that the protein alphabet consists of 20 letters. Each amino acid is defined by a three-nucleotide sequence called the triplet codon.
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- 14.3: Prokaryotic Transcription
- The prokaryotes, which include bacteria and archaea, are mostly single-celled organisms that, by definition, lack membrane-bound nuclei and other organelles. A bacterial chromosome is a covalently closed circle that, unlike eukaryotic chromosomes, is not organized around histone proteins. The central region of the cell in which prokaryotic DNA resides is called the nucleoid. Prokaryotes often have abundant plasmids that are shorter circular DNA molecules that may only contain one or a few genes.
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- 14.4: Eukaryotic Transcription
- Prokaryotes and eukaryotes perform fundamentally the same process of transcription, with a few key differences. The most important difference between prokaryotes and eukaryotes is the latter’s membrane-bound nucleus and organelles. With the genes bound in a nucleus, the eukaryotic cell must be able to transport its mRNA to the cytoplasm and must protect its mRNA from degrading before it is translated.
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- 14.6: Ribosomes and Protein Synthesis
- The synthesis of proteins consumes more of a cell’s energy than any other metabolic process. In turn, proteins account for more mass than any other component of living organisms (other than water), and proteins perform virtually every function of a cell. The process of translation, or protein synthesis, involves the decoding of an mRNA message into a polypeptide product. Amino acids are covalently bonded by interlinking peptide bonds in lengths ranging from ~50 amino acid residues to >1,000.
Thumbnail Image Credit: "DNA Helix String" by qimono , Pixabay is licensed under CC BY 4.0
Contributors and Attributions
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Connie Rye (East Mississippi Community College), Robert Wise (University of Wisconsin, Oshkosh), Vladimir Jurukovski (Suffolk County Community College), Jean DeSaix (University of North Carolina at Chapel Hill), Jung Choi (Georgia Institute of Technology), Yael Avissar (Rhode Island College) among other contributing authors. Original content by OpenStax (CC BY 4.0; Download for free at<EMAIL_ADDRESS>).


6.9: Stock Splits
A stock split is when a corporation reduces the par value of each share of stock outstanding and issues a proportionate number of additional shares. This does affect the number of shares outstanding and, therefore, the number of shares dividends will be paid on. It also may affect the par value and market price per share, reducing them proportionately. However, the total dollar value of the shares outstanding does not change. No journal entry is required for a stock split.
A company has 10,000 shares outstanding. The par value is $16 per share. The fair market value per share is $20. The total capitalization (value of the shares outstanding) is $200,000 (10,000 x $20).
The company declares a 4-for-1 stock split. Multiply the number of shares by 4: there are 40,000 shares outstanding after the split. Divide the par value by 4: each share has a par value of $4 after the split. Also divide the market value per share by 4, resulting in $5 per share. The total capitalization (value of the shares outstanding) is still $200,000 (40,000 x $5).


5.7.0: Review Questions
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- Last updated
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Review Questions
1 .
What is the term used to refer to the integration of an individual’s cultural beliefs into their health care?
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cultural integrity
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culturally competent care
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holistic care
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integrative care
2 .
A patient is admitted to the medical-surgical floor for uncontrolled hypertension. He is a seventy-five-year-old Hispanic man who speaks fluent English. He tells the nurse that he has been seeing a curandero, or traditional healer, for his health issues for the last several years. What is the best initial response from the nurse?
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Ask the patient for a list of all herbs, plants, and special diets that he is currently taking.
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Educate the patient on why adherence to a Western medical treatment plan is better for his health.
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Inform the patient that the treatment he has been receiving from the curandero is not evidence based.
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Tell the patient he is welcome to continue whatever traditional treatments he likes while he is in the hospital.
3 .
 You decide to journal after one of your shifts to reflect on any possible instances of implicit bias in your workday. What might an interaction reflecting implicit bias look like?
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You are frustrated when you learn during report that one of your patients is an 82-year-old male with dementia who is forgetful, difficult to redirect, and often agitated.
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You learn that one of your patients went to another high school in your hometown.
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One of your patients is a 59-year-old female from Syria. During report, the patient’s previous nurse explains to you that the patient’s husband is at the bedside and that when the patient is questioned, he usually answers the questions for her. The nurse states “I think it’s really sexist, but they are from the Middle East.”
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You enter the room of your patient who is a 23-year-old Black female with a history of alcohol misuse. She is on alcohol withdrawal protocol and has orders for frequent vital signs. She shares with you her drinking history and how difficult it is for her to stop. You empathize with her and tell her you are supportive of her efforts to get sober.
4 .
What is the term used to describe nonmedical factors that influence health outcomes, including conditions in which people are born, grow, work, live, and age, and the wider sets of forces and systems shaping the conditions of daily life?
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environmental influences
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life circumstances
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situational occurrences
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social determinants of health
5 .
A process where the patient and nurse seek a mutually acceptable way to deal with competing interests of nursing care, prescribed medical care, and the patient’s cultural needs is known as cultural negotiation. What would be an example that demonstrates cultural negotiation?
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a nurse insisting the patient shave his facial hair
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a nurse demanding daily medication be taken at 0900, when the patient wakes daily at 0500
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a nurse planning to keep the patient’s hijab in place for a surgical procedure
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a nurse requesting a special religious healer visit the patient after admission
6 .
While learning about Chinese culture and attending local presentations on culturally competent care, you discover a local event happening in your area. You decide to ask a group of coworkers to join you in attending a local Chinese New Year parade. This is an example of engaging in what type of cultural diversity practice?
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active learning
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interpersonal awareness
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intrapersonal awareness
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willingness to change
7 .
What is an example that illustrates being culturally competent?
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attending an Indian dance night at a local community center
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educating the patient on your own cultural practices
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implementing a system to improve cardiovascular health screening procedures for Black women at your hospital
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insisting a patient answer your questions instead of her husband answering
8 .
What is an example of cultural humility?
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a nurse allowing a non-English-speaking patient’s son to translate for them
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a nurse listening to a patient who is pregnant explain why they want a traditional midwife from their own culture present in the room while they are delivering their baby
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a nurse administering a new medication to a patient with limited English, even though the translator is late, so that it can be given on schedule
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a nurse giving the patient discharge paperwork printed in their preferred language because they do not have time to verbally discuss discharge instructions with the patient
9 .
What is the purpose of the Theory of Culture Care: Diversity and Universality?
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to provide a framework for transcultural nursing and the development and practice of culturally competent nursing care
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to explain the many cultural differences that exist among communities
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to shape the beliefs of nurses regarding certain health- and wellness-related customs among historically marginalized communities
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to provide patients with a way to address health disparities and systemic inequality in the healthcare system
10 .
Taking a test to examine your own implicit biases is an example of which of the twelve standards of practice for culturally competent nursing care?
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critical reflection
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culturally competent practice
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knowledge of cultures
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social justice
11 .
According to Giger and Davidhizar’s Six Cultural Phenomena, all communication exists in the context of which factor?
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biology
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environment
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space
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time
12 .
What is one important purpose of cultural competency?
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to improve the quality of care leading to better health outcomes for culturally diverse patients
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to ensure that the predominant cultural voices in a community are the ones who shape healthcare practices
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to prevent unscientific beliefs from influencing important medical decisions
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to provide education to cultural communities on how they can best adapt their practices to suit the modern healthcare system
13 .
What term describes the state in which everyone has a fair and just opportunity to obtain their highest level of health?
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health equality
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health equity
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fairness
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justice
14 .
You receive report from another nurse who tells you that your patient is an 82-year-old Chinese female. You know that you will have a good day with this patient as older Asian women are usually very polite, easy to get along with, and compliant with your requests. What prevents you from recognizing diversity and inclusion?
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cultural application
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cultural blindness
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ethnocentrism
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stereotyping
15 .
You have a patient who is Spanish-speaking only, and you are required to obtain their signature on a consent form. The patient requests that their family member interpret for them. What is the most appropriate response?
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Ask the family member to leave the room and use an official translator.
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Find an official translator to assist with the conversation along with the patient and family member.
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Proceed with allowing the family member to interpret.
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Tell the patient that you are only allowed to use official translators or interpreters.


20.4: Treatments and Nutrition
By the end of this section, you should be able to:
- 20.3.1 Identify drug–food interactions related to musculoskeletal and integumentary disorders.
- 20.3.2 Identify treatments and medications that can cause nutritional deficiencies in clients with musculoskeletal and integumentary disorders.
Drug-Food Interactions in Musculoskeletal and Integumentary System Disorders
Drug-food interactions can occur in individuals with musculoskeletal and integumentary system disorders. The most common medications that often interact with food are bisphosphonates and tetracycline antibiotics.
Bisphosphonates are medications commonly used to treat osteoporosis and other bone-related conditions. Nurses should instruct clients to take these medications on an empty stomach with a full glass of water; avoid consuming any food, beverages (except water), or other medications for at least 30 minutes to an hour after taking the medication; and remain upright for at least 30 minutes after taking these medications because they may irritate the esophagus. Food, especially calcium-rich foods, can interfere with the absorption of bisphosphonates (Ganesan et al., 2023).
Tetracycline antibiotics , such as doxycycline, can form insoluble complexes with calcium found in dairy products and calcium supplements. These complexes reduce the absorption of the antibiotic and decrease its effectiveness. To avoid this interaction, nurses should instruct clients to avoid taking tetracycline antibiotics within 2 hours of ingesting calcium-rich foods or supplements. In addition, nurses should inform pregnant clients that tetracycline staining of teeth may occur in the children of individuals who take this medication during pregnancy (DailyMed, 2018).
Medications That Can Cause Nutritional Deficiencies
Nurses should know that chronic use of certain medications can cause nutritional deficiencies and negatively affect the musculoskeletal system. Not all individuals who take these drugs will develop nutritional deficiencies or musculoskeletal problems, but the risk increases with long-term use. Table 20.9 outlines some of the most common medications that can cause nutritional deficiencies related to the musculoskeletal system.
| Medication/Class | Musculoskeletal Effect | Considerations |
|---|---|---|
| Corticosteroids | Long-term use of corticosteroids, such as prednisone, can lead to decreased calcium absorption and increased calcium excretion, which can result in bone loss and osteoporosis. | These medications can interfere with vitamin D metabolism, further affecting bone health. |
| Antacids and proton pump inhibitors (PPIs) | Long-term use of antacids or PPIs can impair the absorption of calcium, magnesium, and vitamin B 12 , which are important for maintaining bone health. | PPI use may lead to hypomagnesemia and hypocalcemia due to decreased absorption of these minerals; therefore, magnesium and calcium levels may need to be monitored in some individuals. |
| Loop diuretics | Loop diuretics (such as furosemide) can increase urinary excretion of calcium, magnesium, and potassium. Prolonged use of these medications may lead to electrolyte imbalances and negatively affect bone density. | Electrolyte levels should be monitored in clients taking loop diuretics. |
| Anticonvulsant medications | Some anticonvulsant drugs, including phenytoin and phenobarbital, may interfere with the metabolism of vitamin D and calcium, potentially leading to bone loss and an increased risk for fractures. | Nurses should inform clients about the increased risk for fractures and teach ways to modify activities as appropriate. |
In addition to knowing which medications may cause vitamin and nutrient deficiencies, the nurse should be aware that certain medications increase a person’s risk for osteoporosis. These medications include:
- Corticosteroids, which are used to treat inflammation
- Antiseizure medications , which are used to prevent seizures and treat other conditions such as neuropathic pain
- Gonadotropin-releasing hormone agonists , which are used for cancers and some gender-affirming treatments
- Depot medroxyprogesterone acetate , which is a contraceptive injection containing the hormone progestin
- Some medications used to treat cancer (American College of Obstetricians and Gynecologists, 2022)


1.3: Structural Organization of the Human Body
By the end of this section, you will be able to:
- Describe the structure of the human body in terms of six levels of organization
- List the eleven organ systems of the human body and identify at least one organ and one major function of each
Before you begin to study the different structures and functions of the human body, it is helpful to consider its basic architecture; that is, how its smallest parts are assembled into larger structures. It is convenient to consider the structures of the body in terms of fundamental levels of organization that increase in complexity: subatomic particles, atoms, molecules, organelles, cells, tissues, organs, organ systems, organisms and biosphere (Figure 1.3).
The Levels of Organization
To study the chemical level of organization, scientists consider the simplest building blocks of matter: subatomic particles, atoms and molecules. All matter in the universe is composed of one or more unique pure substances called elements, familiar examples of which are hydrogen, oxygen, carbon, nitrogen, calcium, and iron. The smallest unit of any of these pure substances (elements) is an atom. Atoms are made up of subatomic particles such as the proton, electron and neutron. Two or more atoms combine to form a molecule, such as the water molecules, proteins, and sugars found in living things. Molecules are the chemical building blocks of all body structures.
A cell is the smallest independently functioning unit of a living organism. Even bacteria, which are extremely small, independently-living organisms, have a cellular structure. Each bacterium is a single cell. All living structures of human anatomy contain cells, and almost all functions of human physiology are performed in cells or are initiated by cells.
A human cell typically consists of flexible membranes that enclose cytoplasm, a water-based cellular fluid together with a variety of tiny functioning units called organelles .  A tissue is a group of many similar cells (though sometimes composed of a few related types) that work together to perform a specific function. An organ is an anatomically distinct structure of the body composed of two or more tissue types. Each organ performs one or more specific physiological functions. An organ system is a group of organs that work together to perform major functions or meet physiological needs of the body.
This book covers eleven distinct organ systems in the human body (Figure 1.4 and Figure 1.5). Assigning organs to organ systems can be imprecise since organs that “belong” to one system can also have functions integral to another system. In fact, most organs contribute to more than one system.
In this book and throughout your studies of biological sciences, you will often read descriptions related to similarities and differences among biological structures, processes, and health related to a person's biological sex. People often use the words "female" and "male" to describe two different concepts: our sense of gender identity, and our biological sex as determined by our chromosomes, hormones, organs, and other physical characteristics. For some people, gender identity is different from biological sex or their sex assigned at birth.
The organism level is the highest level of organization. An organism is a living being that has a cellular structure and that can independently perform all physiologic functions necessary for life. In multicellular organisms, including humans, all cells, tissues, organs, and organ systems of the body work together to maintain the life and health of the organism.


8: Finance Applications
Learning Objectives
In this chapter, you will learn to:
- Solve financial problems that involve simple interest.
- Solve problems involving compound interest.
- Find the future value of an annuity, and the amount of payments to a sinking fund.
- Find the future value of an annuity, and an installment payment on a loan.
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- 8.1: Simple Interest and Discount
- It costs to borrow money. The rent one pays for the use of money is called the interest. The amount of money that is being borrowed or loaned is called the principal or present value. Simple interest is paid only on the original amount borrowed. When the money is loaned out, the person who borrows the money generally pays a fixed rate of interest on the principal for the time period he keeps the money.


10: Shape
The Visual Element of Shape can be natural or man-made, regular or irregular, flat (two-dimensional) or solid (three-dimensional), representational or abstract, geometric or organic, transparent or opaque, positive or negative, decorative or symbolic, colored, patterned or textured. Most of the art we see is two-dimensional: a drawing, a painting, a print, or a photograph which is usually viewed as a flat surface. Most two-dimensional art tries to create the illusion of three dimensions by combining the visual elements to a greater or lesser degree.
In Escher's lithograph, the artist is playing with the illusion of two and three-dimensions in the same image. From an interlocking pattern drawn on a page of his sketchbook, the flat outlined shapes of the reptiles are brought to life by the addition of tone. They step out of their two-dimensional world into a three dimensional landscape of solidly rendered objects that have been selected for their variety of shapes and textures. After a short journey exploring this new environment they return to their original format by losing their tone and adopting their former position within the design - an illusion of a return trip between two and three dimensions.
In general, when we look at shapes we categorize them into organic shapes and geometric shapes. Organic shapes will typically have curved edges or contours whereas geometric shapes will have planes, ellipses, or spherical shapes. When drawing or painting shapes with planes, knowledge of linear perspective is necessary to accurately draw the subject. The angles and curves of shapes appear to change depending on our viewpoint. Our viewpoint is our eye level which (with the exception of an aerial view) is equal to the horizon line used in drawing (linear/vanishing point) perspective.
Shapes used in the visual field can invoke an aesthetic response such as calm, uneasy, or agitated. For example, a triangle has a solid base and planes that point upward. This is a very stable shape to use in a composition. Leonardo Da Vinci used this to great affect in his paintings and Raphael, ever the borrower of great artistic ideas, also begins to use this compositional technique in his paintings as well. Certain shapes have meanings depending upon where they are placed in the visual field. Placing a circle in the upper part of the field may appear to represent the sun or moon. Squares or rectangles may represent solid structures like houses or other buildings. They, too, appear stable and solid. Shapes can also be a form of visual communication as in the octagonal shape of a stop sign, or the inverted triangle used in a yield sign.
Combining shapes into balanced compositions is a worthwhile exercise best started using simple geometric shapes. Even representational subjects can be reduced down to the most basic of shapes. This process is termed abstraction, with the subsequent art called abstract art. Many students feel they do not understand or appreciate abstract art. Think of it this way. When writing and publishing a scientific research article, the author writes a shortened summary of the main points of the paper. This is called an abstract. Artists do the same thing. They simplify and reduce the visual subject pulling connections they see and wish to include in the artwork. Of course there are other forms of abstract art, but looking at it this way may be helpful. Two nice early examples of the process of abstraction of representational subjects into more geometric shapes are found in the work of Theo Van Doseburg and Henri Matisse.
Assignment
Part 1:
Use drawing, painting and/or cut paper to create a “legible, clean and harmonious” abstract geometric design. Use black and white with only one carefully chosen “spot” color. This “spot” color should be used in one area or several specific parts of the design for balance, emphasis or other design consideration.
Part 2:
On a separate sheet of paper list the design considerations incorporated into Part 1. The design elements are from previous chapters and should be typed and double spaced. Be prepared to defend during critique your list relative to your design.
Part 3 (Optional):
Add an organic shape to the design you created in Part 1 or a similar design. The organic shape should be in the form of a drawing, painting or original photograph. Such as the work by Georgia O’Keeffe.


4.1: Description and Objectives
Description
This module covers cybersecurity critical infrastructure and risk management. It introduces the NIST Cybersecurity Framework, the structure of the framework, and how it is used. It also describes the processes of risk management in the framework—framework basics, structure, and a business process management approach to implementing and applying the framework.
Objectives
- Describe basic security service principles (confidentiality, integrity, availability, and authentication) and their relative importance to CI systems.
- Explain basic risk management principles.
- Identify various risk management frameworks and standards, such as the NIST Cybersecurity Framework and the North American Electricity Reliability Council (NERC).
- Describe how to use the framework core process.
- Describe how to use the Framework Implementation Tiers to identify cybersecurity risk and the processes necessary to effectively manage that risk.
- Describe the Cybersecurity Framework Assessment Process Model.
- Demonstrate an understanding of how the framework process holistically manages risk.


7.2.3: Advent Calendar (Usable with Workarounds)
Overall Usability
Usable with Workarounds:
- The activity requires significant adjustments by content authors.
Important Reminders
Use workarounds and best practices for content authoring, including:
- Add a door image with good color contrast
- Hide door borders and frames (the options do not have good color contrast)
- Use a background image that allows for good color contrast
- Do not set audio or video content to auto-play.
Users who require magnification up to 400% may need to be provided with an alternative activity.
Visible Indication of Focus
The Calendar Door buttons, the Close Window button (inside the door), and the contents of the Windows have a visible indication of focus.
Color Contrast
The visible indication of focus has good contrast. The visible indication of focus is a thin black line. Users who require a visible indication of focus with more contrast may use assistive technology.
Keyboard Usability
The activity is usable with the keyboard. To use the activity with the keyboard, inside the Window, from the Close Window button, users should move backwards (Shift + Tab) to cycle through the contents of the Window.
No Keyboard Traps
There are no keyboard traps.
Magnification
Content magnifies up to 200% using browser settings without loss of content or functionality. The calendar content and video content inside the calendar doors does not magnify up to 400%.
An alternative version of the content should be included that supports magnification up to 400% without requiring horizontal scrolling to view content and interact with content.
Usability with Screen Reading Software
The activity is usable with screen reading software. Inside the Window, from the Close Window button, users should move backwards (Shift + Tab) to cycle through the focusable contents of the Window (links and buttons). Users of assistive technology should move the focus to the beginning of the Window contents, then read forward.