14: 20th Century Modern Art Movements- Cubism
Cubism
https://www.metmuseum.org/toah/hd/cube/hd_cube.htm
https://www.tate.org.uk/art/art-terms/c/cubism
Cubism was one of the most influential visual art styles of the early twentieth century. It was created by Pablo Picasso (Spanish, 1881–1973) and Georges Braque (French, 1882–1963) in Paris between 1907 and 1914. The French art critic Louis Vauxcelles coined the term Cubism after seeing the landscapes Braque had painted in 1908 at L’Estaque in emulation of Cézanne . Vauxcelles called the geometric forms in the highly abstracted works “cubes.” Other influences on early Cubism have been linked to Primitivism and non-Western sources. The stylization and distortion of Picasso’s groundbreaking Les Demoiselles d’Avignon ( Museum of Modern Art, New York ), painted in 1907, came from African art. Picasso had first seen African art when, in May or June 1907, he visited the ethnographic museum in the Palais du Trocadéro in Paris.
The Cubist painters rejected the inherited concept that art should copy nature, or that artists should adopt the traditional techniques of perspective, modeling, and foreshortening. They wanted instead to emphasize the two-dimensionality of the canvas. So they reduced and fractured objects into geometric forms, and then realigned these within a shallow, relief-like space. They also used multiple or contrasting vantage points.
Abstraction
Cubism is a form of abstraction. Many students when asked the question "do you understand abstract art?" answer that they do not. A second question is then posed, "do you like abstract art?" and the same students often answer no as well. I explain that like many things in life, if you study the subject and have a general understanding of the "Ws," (who, what, where, when, and why) you may come to appreciate it. That certainly pertains to modern art of the 20th Century. Why was Dada art created? Without the context of the First World War as a backdrop, we may not understand at all this approach to art. However, with the context of the time considered, it makes begins to make sense.
Abstraction in art can be a simplification. Consider the large published scientific article. With what does it begin? An Abstract that condenses the article into the most salient points. It can also be so with abstract art where the artist sees the world and abstracts from reality presenting an imaginative vision of that reality. We have the need to be creative, and when creating art that records only what we see we become bored. Why create what we see? We can use our eyes for that. The Ancient Greeks worked hard to create realistic, life-size images of Koure and Kouros (idealized men and women). They reached that goal with the landmark sculpture Kritios Boy, yet, in about a generation, they began lawfully distorting the human body by evidence of the Riace bronzes and other works of that period. The Riace bronzes go beyond human through idealization and subtle exaggeration. The stylistic formula for the Riace bronzes and much of the Greek Classical and Hellenistic sculpture is attributed to Polykleitos and Pheidias.
http://www.getty.edu/art/collection/artists/3266/polykleitos-greek-active-460-415-bc/
https://www.theacropolismuseum.gr/en/content/kritios-boy-and-severe-style
https://www.moma.org/interactives/exhibitions/2012/inventingabstraction/?work=207
https://www.tate.org.uk/art/artists/theo-van-doesburg-1017
Theo van Doesburg , Composition VIII (The Cow) , 1918, oil on canvas, 37.5 x 63.5 cm, ( MoMA )
Theo van Doesburg’s Composition VIII consists of just fourteen rectangles. One nearly perfect yellow square in the center is surrounded by thirteen rectangles of different proportions: four green, three blue, four black, and two red. They are scattered evenly across the surface, not overlapping (although two pairs abut), and are carefully aligned with the edges of the rectangular white canvas. This seems to be a good, and fairly early, example of what is called “abstract art,” art without any subject matter or visual resemblance to actual objects in nature.
In what appears to be a ludicrous parenthetical statement, however, the title also declares the work to be a cow. And indeed, there are a number of preliminary studies by van Doesburg that may justify that subtitle.
Top left: Theo van Doesburg , Study for Composition (The Cow) , 1917, pencil on paper, 11.7 x 15.9 cm ( MoMA ); top right: Theo van Doesburg , Study for Composition (The Cow) , 1917, pencil on paper, 11.7 x 15.9 cm ( MoMA ); bottom: Theo van Doesburg , Composition (The Cow) , 1917, gouache, oil, and charcoal on paper, 39.4 x 58.4 cm ( MoMA ).
One graphite drawing is immediately recognizable as a cow or bull grazing head down in a field, despite its sketchy execution and the simplification of its muscular contours. Another graphite drawing squares off those contours; the cow’s hip and thigh muscles are simplified into a block, and the rear leg is straightened into an elongated peg. A painted study continues this squaring off process. The head is now composed of a few rectangles: a broad forehead in purple, a black snout terminating in a nested red band, and a yellow antenna-like vertical suggesting an ear.
Now it is possible to see Composition VIII as a cow as well. The central yellow square is a massive, weighty ribcage, and the black rectangle on the left is a hip, with the blue and red below representing two legs. In the lower right is a tripartite head: blue forehead, black snout, and red mouth/nostrils. The title is appropriate after all; at first sight what we see is just an abstract “composition” of colored rectangles, but hidden within is a cow.
The above highlighted areas by authored By Dr. Charles Cramer and Dr. Kim Grant (full article accessed by using link below)
Visualizing Cubism
There are several helpful ways to look at creating an imaginative abstract drawing inspired by Cubism. Imagine that the form you wish to draw is simplified into facets, planes and surfaces each catching light differently. Each may have its own value. Gradations in tones may be added and contrast, by allow the lighter side of the facet, or plane, to be placed next to a darker side of a plane. This creates a shimmering effect like a mosaic, or a cut gemstone. Below are examples of cut gemstones and you will see the light effects on the many facets. Each facet is on a plane catching light in a unique fashion.
Another way of looking at Cubism is to consider what happens when a mirror is cracked and you look at your subject in that mirror. The image is distorted in a way that separates the whole image into shapes shifted in various directions creating a unique image quite similar to the drawings and paintings we sometimes see in Cubism.
Often students have difficulty applying the concepts and new ways of seeing the world that Cubism affords. The first drawing below is the beginning of a Cubist student drawing. It represents an abstract bunny that is broken up into geometric shapes using line and some tone. The student is unsure what to do next and how to complete the drawing. Since this is from an online drawing class I am not able to easily demonstrate drawing solutions using tracing paper and a pencil over the student drawing. Instead I am relying on Photoshop to teach the student how to take this drawing to the next level. Each time I add an element the composition is changed. When adding shapes, values, and textures, the balance is effected and so I work over the entire piece adding elements with the goal of creating a balanced drawing with contrast, movement, and rhythm. Note that I place several values on identified planes, create facets with gradations, patterns with lines, texture with a rough stipple, and areas shadows. Most importantly, I add black shapes which immediately change the look of the work from a rather flat art with middle grays, to a work of art with a full range of values. It becomes more dynamic and complete. I could also play up the use of white in the drawing, but decided to stop where I was considering the art completed to an adequate point. Sometimes it is good to know when to stop when creating art. You can overwork a piece.
Before After
Assignment
• Study the Cubist work of Picasso and Braque
• Be inspired by their “new ways of seeing”
Thumbnail your concept
AFTER YOU DETERMINE YOUR CONCEPT
• Share your concept with the instructor
• Gather your materials
• Carefully execute the design (collage or drawing)
Student Work


8.4: Rules of Replacement
The final rules are fairly simple (in fact, they’re simpler than the rules of implication), but work in a very different way than the rules of implication. They’re called the rules of replacement because they allow you to simply transform or replace a formula (or subformula) with a logically equivalent formula.
If you look above at the rules of implication, you’ll notice that they have a sort of “premise-conclusion” format. If you find the premises, then you can derive the conclusion.
Rules of replacement have a different structure because each is simply a statement that two formulas are logically equivalent. So instead of going in one direction, we can move in either direction in the rule. Here’s an almost ridiculously simple rule of replacement:
~~P :: P
The first thing to note is that little square of dots. That’s what’s called a “metalogical” symbol, but you don’t have to worry too much about that. What it means for our purposes is simple: you can replace these with each other (i.e. 1 they are logically equivalent). A :: B means “you can replace A with B and B with A. The first difference between rules of replacement and rules of implication, therefore, is that rules of replacement are bidirectional —one can move from the left to the right or from the right to left. With the rules of implication, we were only allowed to move down from the premises to the conclusion—we can’t move up from the conclusion back to the premises. That wouldn’t be a valid move. The rules of replacement work in both ways.
This means we can derive from any formula the double-negated form of that formula and vice versa. Put another way, we can remove two negations or add two negations to any formula we want. See how simple some of these rules can be?
Let’s introduce the rest of the rules, and then talk a bit more about differences between rules of replacement and rules of implication. I use Greek Letters Delta , Ohm , and Phi to help get us into the habit of thinking of these as general patterns rather than as specific formulas with just single letters in place of the disjuncts, conjuncts, antecedents, consequents, etc.
\( \neg ( \Delta \bullet \Omega) :: (\neg \Delta \vee \neg \Omega)\)
\( \neg ( \Delta \vee \Omega) :: (\neg \Delta \bullet \neg \Omega)\)
In short: “distributing” a negation the way one might distribute an exponent in math (or, for that matter “undsitributing” as one does moving from right to left in these rules) results in flipping an “and” to an “or” and vice versa.
\( ( \Delta \vee \Omega) :: (\Omega \vee \Delta)\)
\( ( \Delta \bullet \Omega) :: (\Omega \bullet \Delta )\)
In short: you can switch conjuncts with one another and disjuncts with one another. Note: only works with conjunction (\(\bullet\) or \(\wedge\)) and disjunction (\(\vee\)). It doesn’t work with any other connectives.
\([ ( \Delta \vee (\Omega \vee \Phi )] :: [(\Delta \vee \Omega ) \vee \Phi ]\)
\([ ( \Delta \bullet ( \Omega \bullet \Phi )] :: [(\Delta \bullet \Omega ) \bullet \Phi ]\)
In short: if you have two adjacent disjunctions or conjunctions, you may move the parentheses to associate the two letters that are not being associated in the base formula. Note: only works with conjunction ( \(\bullet\) or \(\wedge\)) and disjunction (\(\vee\)). It doesn’t work with any other connectives.
\( \neg \neg \Phi :: \Phi \)
In short: you can add or remove two immediately adjacent negations to a formula or subformula. Note: negations cannot be separated, so this doesn’t work: ~(~Z\(\rightarrow\)X) // (Z\(\rightarrow\)X) [invalid!]
\(( \Phi \rightarrow \Omega) :: (\neg \Omega \rightarrow \neg \Phi ) \)
In short: you can switch antecedent and consequent as long as you add one negation to each or take away one negation from each. Note: must be combined with DN if you want to add one negation and take one negation away using Trans.
\(( \Phi \rightarrow \Omega ) :: (\neg \Phi \vee \Omega )\)
In short: you can change an implication into a disjunction as long as you negate the antecedent. You can also replace a disjunction with at least one negated formula with an implication, so long as the left disjunct is negated. If you do, removed one negation from the left disjunct.
Another important difference between rules of replacement and rules of implication is that rules of replacement can be used on subformulas . A subformula is any formula that is part of another formula. “(A\(\rightarrow\)B)” is a formula, so it’s a subformula of #1 below. “P \(\leftrightarrow\) \(\neg\)” is not a well-formed formula, so it’s not a subformula of #3 below. I can apply a rule of replacement to only part of a formula. So the following derivation is deductively valid even though all of the changes happen to only part of the premise and subsequent formulas:
1. P \(\leftrightarrow\) (A\(\rightarrow\)B) / P \(\leftrightarrow\) \(\neg\)(A \(\wedge\) \(\neg\)B)
2. P \(\leftrightarrow\) (\(\neg\)A\(\vee\)B) 1, Impl
3. P \(\leftrightarrow\) \(\neg\)\(\neg\)(\(\neg\)A \(\vee\) B) 2, DN
4. P \(\leftrightarrow\) \(\neg\)(\(\neg\)\(\neg\)A \(\wedge\)\(\neg\)B) 3, DM
5. P \(\leftrightarrow\) \(\neg\)(A \(\wedge\) \(\neg\)B) 4, DN
If we know that each of the replaced subformulas are logically equivalent, then (since equivalence is transitive), we also know that all of them will be materially equivalent (biconditional) to P (if, that is, proposition 1 is true).
To summarize: rules of replacement are bidirectional and can be applied to subformulas (parts of formulas); whereas rules of implication are unidirectional and apply only to whole Well-Formed Formulas/WFFs.
Stacking rules
Let’s discuss another revision: your instructor may be okay with you stacking a couple of rules on one line. For instance, you may be able to throw a Trans and DN together to add a negation to one part and remove one from another. The important caveat is that you must cite the rules in order to minimize confusion. The above proof might, for instance, look like this:
1. P \(\leftrightarrow\) (A\(\rightarrow\)B) / P \(\leftrightarrow\) \(\neg\)(A \(\wedge\) \(\neg\)B)
2. P \(\leftrightarrow\) \(\neg\)\(\neg\)(\(\neg\)A \(\vee\) B) 1, Impl, DN
3. P \(\leftrightarrow\) \(\neg\)(A \(\wedge\) \(\neg\)B) 2, DM, DN
Generally, this is restricted to Com and DN since these are quick and simple changes rather than dramatic transformations of the formulas involved.
Com and the Strictness of Rules
Once we have these rules on our tool belts, we can actually go back and be a bit stricter with some of the previous rules. Sometimes, your instructor may restrict the rules to apply only to a very specific pattern. Whereas other instructors will be more liberal with the rules. For instance, DS looks like this:
\[\begin{align*}{} & P \vee Q \\ & \underline{\neg P \ \ \ \ } \\ & Q \end{align*}\]
Notice how it’s the left disjunct being negated here. If we left the rule at that, we’d have to use Commutativity, if they were reversed, to switch around the P and Q before we do DS. So it’d have to look like this:
1. Q \(\vee\) P
2. \(\neg\)P /Q
3. P \(\vee\) Q 1, Com
4. Q 2,3 DS
In this case if you stacked rules, you’d need to cite “Com” as the first rule because DS requires Com to happen first .
Similarly with simplification, which sometimes is written to only allow you to simplify the left conjunct. Here’s an example of a proof that follows this rule strictly:
1. P \(\bullet\) ~Q
2. R \(\supset\) Q / ~R \(\bullet\) P
3. P 1, Simp
4. ~Q 1, Com, Simp
5. ~R \(\vee\) Q 2, Impl
6. Q \(\vee\) ~R 5, Com
7. ~R 4, 6, DS
8. ~R \(\bullet\) P 3, 7, Conj
All 14 Rules (simplified form)
Table \(\PageIndex{1}\): All 14 rules (simplified form).
|
Modus Ponens MP |
\[\begin{align*}{} & \Delta \rightarrow \Omega \\ & \underline{\Delta \ \ \ \ \ }\\ & \Omega \end{align*}\] |
Constructive Dilemma CD |
\[\begin{align*}{} & (\varphi \rightarrow \Phi) \wedge (\Delta \rightarrow \Omega) \\ & \underline{(\varphi \vee \Delta) \ \ \ \ \ }\\ & (\Phi \vee \Omega) \end{align*}\] |
|
Modus Tollens MT |
\[\begin{align*}{} & \Delta \rightarrow \Omega \\ & \underline{\neg \Omega \ \ \ \ \ }\\ & \neg \Delta \end{align*}\] |
Simplification simp |
\[\begin{align*}{} & \underline{\Delta \wedge \Omega \ \ }\\ & \Delta \end{align*}\] |
|
Disjunctive Syllogism DS |
\[\begin{align*}{} & \Delta \vee \Omega \\ & \underline{\neg \Omega \ \ \ \ \ }\\ & \Delta \end{align*}\] |
Conjunction conj |
\[\begin{align*}{} & \Phi \\ & \underline{\Omega \ \ \ \ \ }\\ & (\Phi \wedge \Omega) \end{align*}\] |
|
Hypothetical Syllogism HS |
\[\begin{align*}{} & \Delta \rightarrow \Omega \\ & \underline{\Omega \rightarrow \varphi \ \ \ }\\ & \Delta \rightarrow \varphi \end{align*}\] |
Addition add |
\[\begin{align*}{} & \underline{\Phi \ \ \ \ \ }\\ & (\Phi \vee \Omega) \end{align*}\] |
|
DeMorgan's Rules DM |
\( \neg ( \Delta \bullet \Omega) :: (\neg \Delta \vee \neg \Omega)\) \( \neg ( \Delta \vee \Omega) :: (\neg \Delta \bullet \neg \Omega)\) |
Double Negation DN |
\( \neg \neg \Phi :: \Phi \) |
|
Commutativity Com |
\( ( \Delta \vee \Omega) :: (\Omega \vee \Delta)\) \( ( \Delta \bullet \Omega) :: (\Omega \bullet \Delta )\) |
Transposition Trans |
\(( \Phi \rightarrow \Omega) :: (\neg \Omega \rightarrow \neg \Phi ) \) |
| Associativity (Assoc) |
\([ ( \Delta \vee (\Omega \vee \Phi )] :: [(\Delta \vee \Omega ) \vee \Phi ]\) \([ ( \Delta \bullet ( \Omega \bullet \Phi )] :: [(\Delta \bullet \Omega ) \bullet \Phi ]\) |
Material Implication Impl |
\(( \Phi \rightarrow \Omega ) :: (\neg \Phi \vee \Omega )\) |
All 14 Rules (in simplified phrases, not strictly worded)
Table \(\PageIndex{2}\): All 14 rules (in simplified phrases, not strictly worded).
|
Modus Ponens MP |
Find the antecedent, write the consequent |
Constructive Dilemma CD |
Like two MPs side-by-side |
|
Modus Tollens MT |
Negate the consequent, negate the antecedent |
Simplification simp |
Write a conjunct by itself |
|
Disjunctive Syllogism DS |
Negate one disjunct, write the other |
Conjunction conj |
Stick two formulas together with a ‘\(\bullet\)’ (or ‘\(\wedge\)’) between |
|
Hypothetical Syllogism HS |
Take out the “middle man” |
Addition add |
Add anything to the right of a ‘\(\vee\)’ |
|
DeMorgan's Rules DM |
Distribute or un-distribute a negation, change \(\vee\) to \(\wedge\) and vice-versa |
Double Negation DN |
Add or remove two negations |
|
Commutativity Com |
Switch conjuncts or disjuncts with their partners |
Transposition Trans |
Like MT but in replacement form |
| Associativity (Assoc) | Move parentheses around two \(\vee\)’s or two \(\bullet\)’s (or \(\wedge\)’s). |
Material Implication Impl |
Switch between \(\rightarrow\) and \(\vee\), negate or un-negate the antecedent |
Now let’s do a couple of proofs involving the rules of replacement.
1. ~(A\(\bullet\)~B)
2. ~(B\(\bullet\)~C) / A \(\rightarrow\) C
If we pay attention to the placement of the A, the B’s, and the C in 1 and 2, we might notice that they’re pretty close to a Hypothetical Syllogism. There’s a B on the right of one and on the left of the other, so there’s a “middle man” that we can eliminate using HS. The result would be the conclusion. So our goal is to get 1 and 2 into the proper format to be able to apply HS.
How do we do that? Well, looking at 1 and 2, we might notice that there’s really only one rule which allows us to transform a negated conjunction. What rule is that??? Look at the table of rules above. I’ll wait. Right-O! DeMorgen’s rules are the only way we know of (in this class) to transform a negated conjunction or disjunction. So let’s just go ahead and apply DM and see what happens. We distribute the outside negation to both conjuncts and then we change the dot to a vee like so:
3. ~A \(\vee\) ~~B 1,DM
4. ~B \(\vee\) ~~C 2,DM
Remember we want to make conditionals. What rule allows us to change a disjunction into a conditional? Yep! Material Implication will do just that. We also don’t want all of those ugly negations, so we just apply double negation to remove them:
5. A \(\rightarrow\) B 3, DN, Impl
6. B \(\rightarrow\) C 4, DN, Impl
What was that last step we decided on waaaaay at the beginning of the proof? Right on. Hypothetical Syllogism to take of the middle man “B” and leave us with our conclusion:
7. A \(\rightarrow\) C 5, 6, HS
QED! (QED stands for the Latin “Quod Erat Demostrandum” or “which is what was to be demonstrated”. So we can say “QED” when we’re done with a proof or demonstration—that is, when we’ve reached the conclusion—as a way of saying “I proved what I was supposed to prove!”)
How about we try another?
Thank you, Sir! May I please have another!
Uh oh, it’s bad news when your students start talking to you like a Private in Boot Camp! Anyways, here’s the problem:
1. S \(\bullet\) ~M
2. (P \(\vee\) R) \(\supset\) M / ~P \(\bullet\) ~R
One way to think about a problem that looks like this is that we have to find ~P and then find ~R and then we can use Conjunction to stick them together to make the conclusion. This won’t really work in this case, though, since the P and R are right next to one another in the same subformula, so we won’t really be able to get them alone—at least not without some heartache. With that in mind, let’s try something different. What if we tried to negate (P \(\vee\) R)? If we did that, we could use a certain rule to turn ~(P \(\vee\) R) into (~P \(\bullet\) ~R). Which rule?
DeMorgen’s Rule!
Yepperino. So now all we have to do is negate the left side of the condition #2. How do we negate the left side of a conditional?
Modus Ponens!
Close, but not quite.
Modus Tollens!
There you go. Remember “Ponens” means affirming and “Tollens” means negating. In this case we want to negate. What else do we need in order to do an MT?
~M
Yes, dear student. You’re being so smart. Now how are we going to get that ~M? To finish the proof out, I’m going to use a stricter version of Simplification according to which one can only simplify the left side of a conjunction. If that’s the case, we’ll need to switch #1 around left to right before simplifying. Which rule allows us to do this? Com:
3. ~M \(\bullet\) S 1, Com
Excellent, now simplify and then Modus Tollens:
4. ~M 3, Simp
5. ~(P \(\vee\) R) 2,4,MT
What happens next? Remember way back to the beginning when we were talking about turning 5 into the conclusion. We’ll use DeMorgen’s Rule:
6. ~P \(\bullet\) ~R 6, DM
[1] Fun Fact: i.e. is an abbreviation of “id est”, which is Latin for “that is”. ‘e.g.’ is an abbreviation of ‘exempli gratia’, which is Latin for “for example”.


7.2: Important Pentoses and Hexoses
- To identify the structures of D-Ribose and D-2-Deoxyribose and describe how they differ from each other.
- To identify the structures of D-glucose, D-galactose, and D-fructose and describe how they differ from each other.
Although a variety of monosaccharides are found in living organisms, two pentoses and three hexoses are particularly abundant.
Pentoses
The two pentoses are D-Ribose and D-2-deoxyribose. Both are aldopentoses.
Figure \(\PageIndex{1}\): Structures of Two Important Pentoses.
Ribose and Deoxyribose
Both aldopentoses are incorporated into larger biomolecules. Ribose is in the biochemical oxidizing agent NAD + and RNA. Deoxyribose is in DNA. In Ribose all OH groups are on the right. The structural difference between the two monosaccharides is that 2-deoxyribose lacks an alcohol (OH) on carbon atom 2, hence the "2-deoxy" in the name.
Hexoses
The three hexoses are D-glucose, D-galactose, and D-fructose (Figure \(\PageIndex{1}\)). Glucose and galactose are both aldohexoses, while fructose is a ketohexose.
Glucose
D-Glucose, generally referred to as simply glucose, is the most abundant sugar found in nature is an aldohexose. Most of the carbohydrates we eat are eventually converted to glucose by hydrolysis. In a series of biochemical reactions, glucose is used to produce energy for our cells. It is also known by three other names: dextrose , from the fact that it rotates plane-polarized light in a clockwise (dextrorotatory) direction; corn sugar because in the United States cornstarch is used in the commercial process that produces glucose from the hydrolysis of starch; and blood sugar because it is the carbohydrate found in the circulatory system of animals. Normal blood sugar values range from 70 to 105 mg glucose/dL plasma, and normal urine may contain anywhere from a trace to 20 mg glucose/dL urine.
Galactose
D-Galactose is also an aldohexose. It is released when lactose, a disaccharide found in milk, is hydrolyzed. The galactose needed by the human body for the synthesis of lactose is obtained by the conversion of D-glucose to D-galactose. Galactose is also an important constituent of the glycolipids that occur in the brain and the myelin sheath of nerve cells. For this reason it is also known as brain sugar . The structure of D-galactose is shown in Figure \(\PageIndex{2}\). Notice that the configuration differs from that of glucose only at the fourth carbon atom.
Fructose
D-Fructose, shown in Figure \(\PageIndex{2}\), is the most abundant ketohexose. Note that from the third through the sixth carbon atoms, its structure is the same as that of glucose. It occurs, along with glucose and sucrose, in honey (which is 40% fructose) and sweet fruits. Fructose (from the Latin fructus , meaning “fruit”) is also referred to as levulose because it has a specific rotation that is strongly levorotatory (−92.4°). It is the sweetest sugar, being 1.7 times sweeter than sucrose, although many nonsugars are several hundred or several thousand times as sweet (Table \(\PageIndex{1}\)).
| Compound | Relative Sweetness |
|---|---|
| lactose | 16 |
| maltose | 32 |
| glucose | 74 |
| sucrose | 100 |
| fructose | 173 |
| aspartame | 18,000 |
| acesulfame K | 20,000 |
| saccharin | 30,000 |
| sucralose | 60,000 |
Although sweetness is commonly associated with mono- and disaccharides, it is not a property found only in sugars. Several other kinds of organic compounds have been synthesized that are far superior as sweetening agents. These so-called high-intensity or artificial sweeteners are useful for people with diabetes or other medical conditions that require them to control their carbohydrate intake. The synthetic compounds are noncaloric or used in such small quantities that they do not add significantly to the caloric value of food.
The first artificial sweetener—saccharin—was discovered by accident in 1879. It is 300 times sweeter than sucrose, but it passes through the body unchanged and thus adds no calories to the diet. After its discovery, saccharin was used until it was banned in the early 1900s. However, during the sugar-short years of World War I, the ban was lifted and was not reinstated at the war’s end. One drawback to the use of saccharin is its bitter, metallic aftertaste. The initial solution to this problem was to combine saccharin with cyclamate, a second artificial sweetener discovered in 1937.
In the 1960s and 1970s, several clinical tests with laboratory animals implicated both cyclamate and saccharin as carcinogenic (cancer-causing) substances. The results from the cyclamate tests were completed first, and cyclamate was banned in the United States in 1969. Then a major study was released in Canada in 1977 indicating that saccharin increased the incidence of bladder cancer in rats. The US Food and Drug Administration (FDA) proposed a ban on saccharin that raised immediate public opposition because saccharin was the only artificial sweetener still available. In response, Congress passed the Saccharin Study and Labeling Act in 1977, permitting the use of saccharin as long as any product containing it was labeled with a consumer warning regarding the possible elevation of the risk of bladder cancer. Today this warning is no longer required; moreover, the FDA is currently reviewing the ban on cyclamate, as 75 additional studies and years of usage in other countries, such as Canada, have failed to show that it has any carcinogenic effect.
A third artificial sweetener, aspartame, was discovered in 1965. This white crystalline compound is about 180 times sweeter than sucrose and has no aftertaste. It was approved for use in 1981 and is used to sweeten a wide variety of foods because it blends well with other food flavors. Aspartame is not used in baked goods, however, because it is not heat stable.
In the body (or when heated), aspartame is initially hydrolyzed to three molecules: the amino acids aspartic acid and phenylalanine and an alcohol methanol. Repeated controversy regarding the safety of aspartame arises partly from the fact that the body metabolizes the released methanol to formaldehyde. It should be noted, though, that a glass of tomato juice has six times as much methanol as a similar amount of a diet soda containing aspartame. The only documented risk connected to aspartame use is for individuals with the genetic disease phenylketonuria (PKU); these individuals lack the enzyme needed to metabolize the phenylalanine released when aspartame is broken down by the body. Because of the danger to people with PKU , all products containing aspartame must carry a warning label.
Acesulfame K, discovered just two years after aspartame (1967), was approved for use in the United States in 1988. It is 200 times sweeter than sugar and, unlike aspartame, is heat stable. It has no lingering aftertaste.
One of the newest artificial sweeteners to gain FDA approval (April 1998) for use in the United States is sucralose , a white crystalline solid approximately 600 times sweeter than sucrose. Sucralose is synthesized from sucrose and has three chlorine atoms substituted for three OH groups. It is noncaloric because it passes through the body unchanged. It can be used in baking because it is heat stable.
Alcohol sugars such as sorbitol and xylitol are made by the reduction reaction of glucose and xylose. Both glucose and xylose are monosaccharides.
Sorbitol is used to sweeten food for diabetics. This alcohol sugar is not absorbed well by the intestines and therefore has minimal effects on the blood levels of glucose and insulin. The aldehyde functional group in glucose gains two hydrogen atoms from H 2 to form sorbitol (a 1 o -alcohol).
Xylitol is used for sugarless gums. The aldehyde functional group in xylose gains two hydrogen atoms from H 2 to form xylitol (a 1 o -alcohol).
All of the extensive clinical studies completed to date have indicated that these artificial sweeteners approved for use in the United States are safe for consumption by healthy individuals in moderate amounts.
Summary
Two pentoses in living organisms are the aldopentoses D-ribose and D-2-deoxyribose.
Three abundant hexoses in living organisms are the aldohexoses D-glucose and D-galactose and the ketohexose D-fructose.


<IP_ADDRESS>: Variations in Size and Number of Genes
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Learning Objectives
- Describe how variations in the size and number of genes can arise through evolutionary mechanisms
Variations in Size and Number of Genes
Genetic diversity refers to any variation in the nucleotides, genes, chromosomes, or whole genomes of organisms. Genetic diversity at its most elementary level is represented by differences in the sequences of nucleotides (adenine, cytosine, guanine, and thymine) that form the DNA (deoxyribonucleic acid) within the cells of the organism. The DNA is contained in the chromosomes present within the cell; some chromosomes are contained within specific organelles in the cell (for example, the chromosomes of mitochondria and chloroplast). Nucleotide variation is measured for discrete sections of the chromosomes, called genes. Thus, each gene compromises a hereditary section of DNA that occupies a specific place of the chromosome, and controls a particular characteristic of an organism.
Chromosomes
Most organisms are diploid, having two sets of chromosomes, and therefore two copies (called alleles ) of each gene. However, some organisms can be haploid, triploid, or tetraploid (having one, three, or four sets of chromosomes respectively). Within any single organism, there may be variation between the two (or more) alleles for each gene. This variation is introduced either through mutation of one of the alleles, or as a result of sexual reproduction.
During sexual reproduction, offspring inherit alleles from both parents and these alleles might be slightly different, especially if there has been migration or hybridization of organisms, so that the parents may come from different populations and gene pools. Also, when the offspring’s chromosomes are copied after fertilization, genes can be exchanged in a process called sexual recombination. Harmless mutations and sexual recombination may allow the evolution of new characteristics.
Genome Size and Number
Genome size is usually measured in base pairs (or bases in single-stranded DNA or RNA). The C-value is another measure of genome size. The C-value refers to the amount, in picograms, of DNA contained within a haploid nucleus (e.g. a gamete) or one half the amount in a diploid somatic cell of a eukaryotic organism. In some cases (notably among diploid organisms), the terms C-value and genome size are used interchangeably, however in polyploids the C-value may represent two or more genomes contained within the same nucleus.
Different species can have different numbers of genes within the entire DNA or genome of the organism. However, a greater total number of genes might not correspond with a greater observable complexity in the anatomy and physiology of the organism (i.e. greater phenotypic complexity). For example, the predicted size of the human genome is not much larger than the genomes of some invertebrates and plants, and may even be smaller than the Indian rice genome. In humans, more proteins are encoded per gene than in other species. In prokaryotic genomes, research has shown that there is a significant positive correlation between the C-value of prokaryotes and the amount of genes that compose the genome. This indicates that gene number is the main factor influencing the size of the prokaryotic genome.
Genes vs Genome Size
In eukaryotic organisms, there is a paradox observed, namely that the number of genes that make up the genome does not correlate with genome size. In other words, the genome size is much larger than would be expected given the total number of protein coding genes. Genome size can increase by duplication, insertion, or polyploidization and the process of recombination can lead to both DNA loss or gain. It is also possible that genomes can shrink due to deletions.
A famous example for such gene decay is the genome of Mycobacterium leprae, the causative agent of leprosy. M.leprae has lost many once-functional genes over time due to the formation of pseudogenes. This is evident in looking at its closest ancestor Mycobacterium tuberculosis. M. leprae lives inside and replicates inside of a host and due to this arrangement it does not have a need for many of the genes it once carried which allowed it to live and prosper outside of the host. Thus over time these genes have lost their function through mechanisms such as mutation causing them to become pseudogenes. It is beneficial to an organism to rid itself of non-essential genes because it makes replicating its DNA much faster and more energy-efficient.
An example of increasing genome size over time is seen in filamentous plant pathogens. These plant pathogen genomes have been growing larger over the years due to repeat-driven expansion. The repeat-rich regions contain genes coding for host interaction proteins. With the addition of more and more repeats to these regions the plants increase the possibility of developing new virulence factors through mutation and other forms of genetic recombination. In this way it is beneficial for these plant pathogens to have larger genomes.
Contributions and Attributions
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Key Points
- Harmless mutations and sexual recombination of chromosomes may allow the evolution of new characteristics.
- Genome size can be affected by various events, including duplication, insertion, recombination, deletion and polyploidization events.
- Genome size can be affected by evolution of an organism and result is an increased or decreased need for specific genes for survival based on behavior.
- The human genome exemplifies the concept that complexity does not always correlate with an increase in genome size; there are fewer protein coding genes present than expected relative to the genome size.
Key Terms
- polyploidization : hybridization that leads to polyploidy
- pseudogene : a segment of DNA that is part of the genome of an organism, and which is similar to a gene but does not code for a gene product
- genome : the cell’s complete genetic information packaged as a double-stranded DNA molecule