15.8: Quality Assurance
What you’ll learn to do: summarize common management techniques used to ensure high-quality goods and services
In this section, you’ll learn about common quality-management techniques used in the production of goods and services. We’ll also discuss the role of national and international quality standards in industry.
Learning Objectives
- Explain the benefits of national and international quality standards in the production of goods and services.
Quality Assurance
What is quality ? According to the American Society for Quality, quality refers to “the characteristics of a product or service that bear on its ability to satisfy stated or implied needs” (American Society of Quality, 2024).
As a customer, you’re constantly assured that when products and services make it to market, they’re of the highest quality. If they aren’t—if they fail to meet your expectations or to live up to claims—you may decide to avoid certain brands or give up on those products/services altogether. When companies can’t deliver quality goods or services, they risk losing trust, loyalty, and business.
Nowhere are the high stakes of quality more evident than in the case of a product recall—when a company requests the return of a product after the discovery of safety issues or product defects that might endanger the consumer.
Consider Samsung’s recent recall of its popular tablet computer, the Galaxy Note 7, in October 2016:
"Samsung has announced an expanded voluntary recall on all original and replacement Galaxy Note 7 devices sold or exchanged in the United States in cooperation with the U.S. Consumer Product Safety Commission and in partnership with carriers and retailers. Since the affected devices can overheat and pose a safety risk, we are asking consumers with a Galaxy Note 7 to power it down and contact the carrier or retail outlet where they purchased their device.” ( Samsung, 2016).
If you bought a Galaxy Note 7, you probably weren’t expecting that it might catch fire during regular use! If you held shares of Samsung in your stock portfolio, you probably weren’t expecting the company’s stock price to plummet practically overnight, either. After all, Samsung was the world's number-one manufacturer of smartphones, and as of September 30, 2016, it had sold 2.5 million Galaxy Note 7 devices worldwide. Yet this is exactly what happened. Besides affecting millions of customers and taking a toll on shareholders, the recall meant lost income for retailers, who had to pull the device from their shelves, and for many of Samsung’s suppliers. This quality debacle impacted the entire supply chain.
Given the devastating financial and, in some cases, legal consequences of selling inferior goods or services, how do companies actually ensure that they’re producing products and offering services that meet customer expectations for quality? We will examine just a few of the ways companies manage the production of quality goods and services.
Statistical Process Control
Statistical process control (SPC) is a quality control method that uses statistical or mathematical methods to monitor and control a process. SPC aims to ensure that production operates at its full potential. “Full potential” indicates the point where the process produces as much conforming product as possible with a minimum (if not the total elimination) of defective parts, rework, or scrap. SPC can be applied to any process in which the product can be measured. Key tools used in SPC include control charts focusing on continuous improvement.
Example \(\PageIndex{1}\)
Margie is the production manager at Wanda’s Widgets. The company uses SPC as its approach to quality assurance. Several times per day, the quality assurance team comes to the production floor and takes a sample of widgets from the production line. These widgets are closely inspected to ensure that they meet company standards. Everything from their weight to the uniformity of the paint is closely inspected and entered into the SPC software program. When the data are analyzed, if the output from the SPC software indicates that the widgets do not meet the standard, Margie is alerted that there is an issue, and production may be stopped until the process produces as many perfect widgets as possible.
Benchmarking
Benchmarking involves comparing one’s business processes and performance metrics to industry bests and best practices from other companies. Dimensions typically measured are quality, time, and cost. In the process of best-practice benchmarking, management identifies the best firms in their industry—or in another industry where similar processes exist—and compares the results and processes of those studied (the “targets”) to one’s own results and processes. In this way, management learns how well the targets perform and, more importantly, the business processes that explain why those firms are successful.
Benchmarking is used to measure performance using a specific indicator (cost per unit of measure, productivity per unit of measure, cycle time of x per unit of measure or defects per unit of measure), resulting in a performance metric compared with others. Benchmarking may be a one-time event but is often treated as a continuous process in which organizations continually seek to improve their practices.
Lean Manufacturing
The central idea of lean manufacturing is actually quite simple: Work relentlessly to eliminate waste from the manufacturing process. In this context, “waste” is defined as any activity that doesn’t add value from the customer’s perspective. Almost every company has a tremendous opportunity to improve using lean manufacturing techniques. Lean principles were developed by the Japanese manufacturing industry—by Toyota and the Toyota Production System (TPS) specifically. Lean manufacturing is based on the following goals and assumptions:
- Continuous improvement
- Respect for people
- A long-term approach to process improvement
- The right process will produce the right results
- Add value to the organization by developing your people and partners
- Continuously solving root problems
Did You Know...?
Toyota originally began sharing the TPS with its parts suppliers in the 1990s . Because of interest in the program from other organizations, Toyota began offering instruction in the methodology to others. Toyota has even “donated” its system to charities, providing its engineering staff and techniques to nonprofits in an effort to increase their efficiency and thus ability to serve people. For example, Toyota assisted the Food Bank For New York City to significantly decrease waiting times at soup kitchens, packing times at a food distribution center, and waiting times in a food pantry ( El-Naggar, 2013) .
Six Sigma
In the United States, another approach to quality management was formulated at Motorola in 1986 and was named Six Sigma (6σ) . Whereas lean management is focused on eliminating waste and ensuring efficiency, Six Sigma focuses on eliminating defects and reducing variability. The following features also set Six Sigma apart from other quality-improvement initiatives:
- A clear focus on achieving measurable and quantifiable financial returns from any Six Sigma project. To determine the financial return on a quality initiative, the cost of quality (COQ) must be calculated. The cost of quality has two parts, added together: the cost of prevention and the cost of failure (or nonconformance). If spending more on prevention reduces the cost of failure by an even greater amount, the total cost of quality is reduced, and such a project would make good business sense.
- An increased emphasis on strong and passionate management leadership and support.
- A clear commitment to making decisions based on verifiable data and statistical methods rather than on assumptions and guesswork.
Six Sigma identifies individuals considered “experts in quality” and awards titles like Champion and Master Black Belt. By the late 1990s, about two-thirds of the top five hundred companies in the United States had begun Six Sigma projects, including Ford, which had allowed its quality programs to slip.
International Quality Standards
As a consumer, wouldn’t you like to know which companies ensure their products meet quality specifications? Or, might you want to know which companies take steps to protect the environment? Some consumers want to know which companies continuously improve their performance in both areas—that is, practice both quality management and environmental management. By the same token, if you were a company doing a good job in these areas, wouldn’t you want potential customers to be aware of your achievements? It might also be worthwhile to find out whether your suppliers were conscientious in these areas—and even your suppliers’ suppliers.
ISO 9000 and ISO 14000
Through the International Organization for Standardization (ISO), a nongovernmental agency based in Switzerland, it’s possible to find out the kind of information just mentioned. The resources of this organization will enable you to identify organizations with people and processes in place for delivering products that satisfy customers’ quality requirements. You can also find out which organizations work to reduce the negative impact of their activities on the environment. Working with representatives from various countries, the organization has established the ISO 9000 family of international standards for quality management and the ISO 14000 family of international standards for environmental management.
ISO standards focus on how a company does its work, not on its output (though there’s certainly a strong correlation between how a business functions and the quality of its products). Compliance with ISO standards is voluntary, and the certification process is time-consuming and complex. Even so, hundreds of thousands of organizations around the world are ISO 9000 and ISO 14000 certified (International Organization for Standardization, 2022). ISO certification has become an internationally recognized symbol of quality management and is increasingly essential to being competitive in the global marketplace.
Malcolm Baldrige National Quality Award
To provide encouragement and a consistent standard, the U.S. government created the Malcolm Baldrige National Quality Award in 1987 to encourage companies to improve quality; the award was named for Malcolm Baldrige, who was the U.S. secretary of commerce from 1981 to 1987 (National Institute of Standards and Technology, 2023). The Commerce Department’s National Institute of Standards and Technology (NIST) manages the Baldrige Award in cooperation with the private sector. An organization may compete for the award in one of six categories: manufacturing, service, small business, health care, education, and nonprofit (including government agencies). An independent board of examiners recommends the Baldrige Award recipients after evaluating them in the following seven areas defined by the Baldrige Excellence Framework:
- leadership
- strategy
- customers
- measurement
- analysis and knowledge management
- workforce
- operations
- results
Past recipients of the Baldridge Award include the following:
- Price-Waterhouse-Coopers Public Sector Practice, McLean, VA
- Pewaukee School District, Pewaukee, WI
- Concordia Publishing House, St. Louis, MO
- City of Irving, Irving, TX
- Lockheed Martin Missiles and Fire Control, Grand Prairie, TX
- Nestlé Purina PetCare Co., St. Louis, MO
"No one knows the cost of a defective product—don’t tell me you do. You know the cost of replacing it, but not the cost of a dissatisfied customer."
—W. Edwards Deming


1.21: SIM Medium Assignment
Name ______________________
Data and Observations
Sulfur Reduction
| Organism (write name out fully) |
Observations (include color and location) |
Result +/- |
Interpretation |
Motility
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Organism (may properly abbreviate) |
Observations |
Result +/- |
Interpretation |
Indole Production
| Organism (write name out fully) |
Observations (include color and location) |
Result +/- |
Interpretation |
Post Lab Questions
- Would this medium be considered selective, differential or both?
2. What is the purpose of comparing your results to an uninoculated inoculated tube?
The inability of this test to distinguish between H2S produced by putrefaction and H2S produced by reduction in the final step of an anaerobic ETS, can be described as poor
3. in this test system. The indole test determines if the organism has the enzyme tryptophanase. The substrate for this enzyme is:
4. What component of the medium supplies this substrate?
Predict what your observations would look like for Citrobacter freundii if you made each of these mistakes in preparing the medium. Explain in one sentence.
5. Used 15 g/L agar
6. Left out the ferrous ammonium sulfate Fe(NH4)2(SO4)
7. Left out sodium thiosulfate Na2S2O3
8. If you left out the sodium thiosulfate would that increase the specificity or sensitivity of the test?


21.2: Coastal Landforms and Processes
A coast or the coastal zone is a dynamic region where land is sculpted and shaped by wave action and currents. Barring the effects of tectonic uplift and sea level change, erosion is the dominate geomorphic process acting on coasts. Coastal sediments are subject to multiple episodes of erosion, transportation and deposition, though a net seaward transport takes place on a global scale. The deep ocean floor becomes the resting place for terrestrial sediment eroded from the land.
The combined effect of waves, currents and tides result in a variety of gradational processes acting in the coastal zone. Most important is abrasion , caused by the scraping or impact of sediment carried by water thrown against shore materials. Breakers are particularly effective at lifting larger rocks and hurling them against the shore.
Hydraulic action caused by the direct impact of waves on the coast can be an effective geomorphic agent. Enormous pressures can build as water and air are compressed into rock fractures. Solution is locally important especially where soluble rock is exposed along the shore.
The water level in the surf zone increases as waves approach shore at an angle. The rising water moves parallel to the shore as a longshore current. Beach drifting transports sand grains along the beach as waves strike the shore at an oblique angle. Sediment is carried landward when water rushes across the beach as swash . Sediment is carried back toward the ocean as backwash . The continual up rush and backwash carries sand in a zig-zag like movement along the shore.
Depositional Coastal Landforms
Beaches
A beach is a deposit of loose sediment adjacent to a body of water. Though sand is common to most beaches, a remarkable diversity of sediment size, from boulders to fine silt is found on beaches around the world. Larger particles and steeper slopes are found where wave action is high. Fine particles and gentle slopes are characteristic of beaches exposed to low wave action.
Most midlatitude beaches undergo a cycle of erosion and deposition following the seasonal changes in wave action. During the winter, midlatitude storms are more vigorous producing more wave action and erosion. Hence, beaches tend to narrow during the winter. Wave action subsides during the summer as storms weaken somewhat favoring deposition over erosion and producing broader beaches.
Spits and bars
A sand spit is one of the most common coastal landforms. A sand spit is a linear accumulation of sediment that is attached to land at one end. Sand carried parallel to shore by longshore drift may eventually extend across a bay or between headlands especially where water is relatively calm. Spits are typically elongate, narrow features built to several dozen feet by wind and waves.
Spits often form when wave energy decreases as a result of wave refraction in a bay. When a coastline turns abruptly, wave energy is dissipated by divergence of wave trajectories, causing sediment to accumulate as the water loses its ability to transport.
Spits can extend across the mouth of a bay, but wave action is usually strong enough to wash sand out to sea or be deposited in the embayment. They may curve into the bay or stretch across connecting to the other side as a baymouth bar . When the bay is closed off by a bar it becomes a lagoon .
Wave energy also dissipates in the lee of large sea stacks or islets. Wave refraction sweeps sediment behind the obstruction from two directions, depositing it as a slender finger called a tombolo .
Barrier Islands
Coastlines paralleled by offshore narrow strips of sand dunes, salt marshes and beaches are known as barrier islands . Barrier island complexes stretch along the southeastern coast of North America from Long Island, New York to the Gulf coast of Texas. Many believe barrier islands originated as offshore bars built by waves breaking on a shallow shore. When waves begin to feel the tug of the ocean floor, they push sand toward shore as they break. The return undertow sweeps sand back to settle on the developing bar. These offshore bars were later exposed when the continent rebounded after ice age glaciers melted.
Larger barrier islands have been sought as "ideal" places to build vacation homes. Such scenic places are susceptible to the destructive forces of nature such as hurricanes, however. In 1989 Hurricane Hugo hit the U.S. mainland just north of Charleston, South Carolina, with sustained winds of 215 kilometers per hour and a storm surge of more than 6 meters. The barrier islands along the South Carolina coast are generally less than 3 meters above sea level. Most islands were inundated with water and in many places, beaches were severely eroded and dunes were leveled. Much of the sand eroded from the beach was over washed landward or carried offshore. South Carolina suffered $5.9 billion in damage and 29 people died.
Erosional Coastal Landforms
Some of the most spectacular scenery is found along coastlines and produced by the effects of wave erosion. Wave erosion undercuts steep shorelines creating coastal cliffs. A sea cliff is a vertical precipice created by waves crashing directly on a steeply inclined slope. Hydraulic action, abrasion, and chemical solution all work to cut a notch at the high water level near the base of the cliff. Constant undercutting and erosion causes the cliffs to retreat landward.
Sea caves form along lines of weakness in cohesive but well-jointed bedrock. Sea caves are prominent headlands where wave refraction attacks the shore.
A sea arch forms when sea caves merge from opposite sides of a headland. If the arch collapses, a pillar of rock remains behind as a sea stack .
Seaward of the retreating cliffs, wave erosion forms a broad erosional platform called a wave-cut bench or wave-cut platform . After the constant grinding and battering, eroded material is transported to adjacent bays to become beaches or seaward coming to rest as a wave-built terrace . If tectonic forces raise the bench above the water level a marine terrace forms. Some shorelines have several marine terraces creating during various episodes of uplift.


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1.1: What is biogeochemistry?
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1.2: Chemistry and Energetics of the Life Process
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About a third of the chemical elements cycle through living organisms, which are responsible for massive deposits of silicon, iron, manganese, sulfur, and carbon. Large quantities of methane and nitrous oxide are introduced into the atmosphere by bacterial action, and plants alone inject about 400,000 tons of volatile substances (including some metals) into the atmosphere annually.
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1.3: Systems, Cycles, Reservoirs, and Fluxes
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Thinking about the Earth as a system has become, in recent years, much in vogue.
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1.4: Gaia - Bioregulation of the Environment
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The physical conditions under which life as we know it can exist encompass a relatively narrow range of temperature, pH, osmotic pressure, and ultraviolet radiation intensity. It seems remarkable enough that life was able to get started at all; it is even more remarkable that it has continued to thrive in the face of all the perils that have, or could have occurred, during the past 3 billion years or so.


9.5: Conclusion
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In this chapter we set out to draw a diagram of religion and culture in world affairs. The aim was to show that religious and cultural factors matter if we want to deepen our understanding of international relations. The method has been to define elements of each concept and consider the impact of these elements on aspects of our individual, national and international experience. Hopefully, you are convinced that understanding religious and cultural issues is necessary if you want to join some of the most important discussions about world politics today. There is little that concerns IR today that does not involve elements of religion or culture, or both. Equally, it is important to recognise as a final thought that we have only just begun to explore these issues and we need to go deeper in our consideration of the importance of religious and cultural actors and interests. Understanding them will help us better understand an ever more complex and divided world.


14.3: Standard Libraries
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Overview
Many common or standard functions, whose definitions have already been written, are ready to be used in any program. They are organized into a group of functions (think of them as several books) and are collectively called a standard library . There are many functions organized into several libraries For example, within most programming languages many math functions exist and have been coded (and placed into libraries). These functions were written by programmers and tested to ensure that they work properly. In most cases, the functions were reviewed by several people to double and triple check to ensure that they did what was expected. We have the advantage of using these functions with confidence that they will work properly in our programs, thus saving us time and money.
Discussion
The main program must establish the existence of functions used in that program. Depending on the programming language, there is a formal way to:
- define a function
- declare a function (a prototype is a declaration to a compiler)
- call a function
When we create functions in our program, we usually see them in the following order in our source code listing:
- declare the function (prototype)
- call the function
- define the function
When we use functions created by others that have been organized into a library, we include a header file in our program which contains the prototypes for the functions. Just like functions that we create, we see them in the following order in our source code listing:
- declaring the function (prototype provided in the include file)
- call the function (with parameter passing of values)
- define the function (it is either defined in the header file or the linker program provides the actual object code from a Standard Library object area)
In most cases, the user can look at the prototype and understand exactly how the communications (parameter passing) into and out of the function will occur when the function is called. Let’s look at the math example of absolute value.
| Language | Example |
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#include <cmath>
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| C# |
Math.Abs(number);
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| Java |
Java.lang.Math.abs(number)
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| JavaScript |
Math.abs(number);
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| Python |
abs(number)
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| Swift |
abs(number)
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Not wanting to have a long function name the designers named it: abs instead of “absolute”. This might seem to violate the identifier naming rule of using meaningful names, however, when identifier names are established for standard libraries they are often shortened to a name that is easily understood by all who would be using them. If I had two integer variables named apple and banana; and I wanted to store the absolute value of banana into apple; then a line of code to call this function would be:
apple = abs(banana);
Let’s say it in English, pass the function absolute the value stored in variable banana and assign the returning value from the function to the variable apple. Thus, if you know the prototype you can usually properly call the function and use its returning value (if it has one) without ever seeing the definition of the code (i.e. the source code that tells the function how to get the answer; that is written by someone else; and either included in the header file or compiled and placed into an object library; and linked during the linking step of the Integrated Development Environment (IDE).
Key Terms
- abs
- A function within a standard library which stands for absolute value.
- confidence
- The reliance that Standard Library functions work properly.
- standard library
- A set of specific task functions that have been added to the programming language for universal use.


Page 10.2: Biotechnology and Genetic Engineering
In the early 1990s, an emerging disease was destroying Hawaii’s production of papaya and threatening to decimate the $11 million industry (Figure \(\PageIndex{1}\)). Fortunately, a man named Dennis Gonsalves, who was raised on a sugar plantation and then became a plant physiologist at Cornell University, would develop papaya plants genetically engineered to resist the deadly virus. By the end of the decade, the Hawaiian papaya industry and the livelihoods of many farmers were saved thanks to the free distribution of Dr. Gonsalves seeds.
Figure \(\PageIndex{1}\). The symptoms of papaya ringspot virus are shown on the tree (a) and fruit (b). “This work” is in the Public Domain, CC0 .The development of a new strain of crop is an example of agricultural biotechnology: a range of tools that include both traditional breeding techniques and more modern lab-based methods. Traditional methods date back thousands of years, whereas biotechnology uses the tools of genetic engineering developed over the last few decades. Genetic engineering is the name for the methods that scientists use to introduce new traits to an organism. This process results in genetically modified organisms, or GMO . For example, plants may be genetically engineered to produce characteristics to enhance the growth or nutritional profile of food crops. GMO that are crop species are commonly called genetically engineered crops , or GE crops for short
The History of Genetic Modification of Crops
Nearly all the fruits and vegetables found in your local market would not occur naturally. In fact, they exist only because of human intervention that began thousands of years ago. Humans created the vast majority of crop species by using traditional breeding practices on naturally-occurring, wild plants. These practices rely upon selective breeding (human assisted-breeding of individuals with desirable traits). Traditional breeding practices, although low-tech and simple to perform, have the practical outcome of modifying an organism’s genetic information, thus producing new traits.
An interesting example is maize (corn). Biologists have discovered that maize was developed from a wild plant called teosinte. Through traditional breeding practices, humans living thousands of years ago in what is now Southern Mexico began selecting for desirable traits until they were able to transform the plant into what is now known as maize. In doing so, they permanently (and unknowingly) altered its genetic instructions, allowing for new traits to emerge. Considering this history, we might ask the question: is there really such a thing as “non-GMO” maize?
Figure \(\PageIndex{2}\). A wild grass called teosinte was genetically modified through selective breeding to produce what is now known as maize (corn). This process of transformation started thousands of years ago by indigenous people of what is now Mexico. “This work” by Nicolle Rager Fuller, National Science Foundation is in the Public Domain, CC0 .This history of genetic modification is common to nearly all crop species. For example, cabbage, broccoli, Brussel sprouts, cauliflower, and kale were all developed from a single species of wild mustard plant (Figure \(\PageIndex{2}\)). Wild nightshade was the source of tomatoes, eggplant, tobacco, and potatoes, the latter developed by humans 7,000 – 10,000 years ago in South America.
Figure \(\PageIndex{3}\). Brassica oleracea is a plant in the mustard family and is known as wild cabbage. From it were developed many familar crops, such as cauliflower, broccoli, Brussel sprouts, and of course, cabbage. “Brassica oleracea wild” by Kulac is licensed under CC BY-SA 4.0Traditional Breeding v. Modern Genetic Engineering
To produce new traits in livestock, pets, crops, or other type of organism, there almost always has to be an underlying change in that organism’s genetic instructions. What many people might not understand is that traditional breeding practices do, in fact, result in permanent genetic changes and is therefore a type of genetic modification. This misunderstanding may arise because traditional breeding practices do not require sophisticated laboratory equipment or any knowledge of genetics, which some may see as a perquisite for genetic modification.
How do traditional breeding practices compare to modern genetic engineering? Both result in changes to an organism’s genetic information, but the magnitude of those changes varies amongst the two techniques (Figure \(\PageIndex{3}\)). Traditional breeding shuffles all of the genes between the two organisms being bred, which can number into the tens of thousands (maize, for example, has 32,000 genes). When mixing such a large number of genes, the results can be unpredictable. Modern genetic engineering is more precise in the sense that biologists can modify just a single gene. Also, genetic engineering can introduce a gene between two distantly-related species, such as inserting a bacterial gene into a plant. Such transfer might seem unusual, but it is not without its equivalent in nature. In a process called horizontal gene transfer , DNA from one species can be inserted into a different species. One recent study, for example, has found that humans contain about 150 genes from other species, including bacteria.
Figure \(\PageIndex{4}\). Both traditional breeding and modern genetic engineering produce genetic modifications. Genetic engineering allows for fewer and more precise genetic modifications. FDA graphic by Michael J. Ermarth (Methods of Plant Breeding) [CC0 Public Domain], via Wikimedia Commons.Potential Benefits of Genetic Engineering
Enhanced nutrition
Advances in biotechnology may provide consumers with foods that are nutritionally-enriched (Figure \(\PageIndex{4}\)), longer-lasting, or that contain lower levels of certain naturally occurring toxins present in some food plants. For example, developers are using biotechnology to try to reduce saturated fats in cooking oils, reduce allergens in foods, and increase disease-fighting nutrients in foods. Biotechnology may also be used to conserve natural resources, enable animals to more effectively use nutrients present in feed, decrease nutrient runoff into rivers and bays, and help meet the increasing world food and land demands.
Figure \(\PageIndex{5}\). White rice and Golden rice. Genetically engineered “Golden Rice” contains up to 35 μg β-carotene per gram of rice, which could prevent millions of cases of blindness worldwide.Cheaper and More Manageable Production
Biotechnology may provide farmers with tools that can make production cheaper and more manageable. For example, some biotechnology crops can be engineered to tolerate specific herbicides, which make weed control simpler and more efficient. Other crops have been engineered to be resistant to specific plant diseases and insect pests, which can make pest control more reliable and effective, and/or can decrease the use of synthetic pesticides. These crop production options can help countries keep pace with demands for food while reducing production costs.
Improved pest control
Biotechnology has helped to make both pest control and weed management safer and easier while safeguarding crops against disease. For example, genetically engineered insect-resistant cotton has allowed for a significant reduction in the use of persistent, synthetic pesticides that may contaminate groundwater and the environment. In terms of improved weed control, herbicide-tolerant soybeans, cotton, and corn enable the use of reduced-risk herbicides that break down more quickly in soil and are non-toxic to wildlife and humans.
Figure \(\PageIndex{6}\). Worldwide, GE crops account for 12% of all agricultural production in 2015. Shown here are GE crops by nation. “This work” by NASEM is in the Public Domain, CC0 .Potential Concerns about Genetically Engineered Crops
The complexity of ecological systems presents considerable challenges for experiments that assess the risks and benefits of GE crops. Assessing such risks is difficult, because both natural and human-modified systems are highly complex and fraught with uncertainties that may not be clarified until long after an experimental introduction has been concluded. Critics of GE crops warn that their cultivation should be carefully considered within broader ecosystems because of their potential benefits and hazards to the environment.
In addition to environmental risks, some people are concerned about potential health risks of GE crops because they feel that genetic modification alters the intrinsic properties, or essence, of an organism. As discussed above, however, it is known that both traditional breeding practices and modern genetic engineering produce permanent genetic modifications. Further, traditional breeding practices actually have a larger and more unpredictable impact on a species’ genetics because of its comparably crude nature. Considering this, it is wise that both new GE crops and traditionally produced crops should be studied for potential human health risks.
To address these various concerns, the US National Academies of Sciences, Engineering, and Medicine (NASEM) published a comprehensive, 500-page report in 2016 that summarized the current scientific knowledge regarding GE crops. The report, titled Genetically Engineered Crops: Experiences and Prospects , reviewed more than 900 research articles, in addition to public comments and expert testimony. Results from this seminal report, hereafter referred to as the “ GE Crop Report ” for brevity, is shared in the various subsections below.
Interbreeding with Native Species
Through interbreeding, or hybridization, GE crops might share their genetically-modified DNA with wild relatives. This could affect the genetics of those wild relatives and have unforeseen consequences on their populations, and could even have implications for the larger ecosystem. For example, if a gene engineered to confer herbicide resistance were to pass from a GE crop to a wild relative, it might transform the wild species into a ‘ super weed’ – a species that could not be controlled by herbicide. Its rampant growth could then displace other wild species and the wildlife that depends on it, thus inflecting ecological harm.
NASEM’s GE Crop Report did find some evidence of genetic transfer between GE crops and wild relatives. However, there was no evidence of ecological harm from that transfer. Clearly, continued monitoring, especially for newly-developed crops, is warranted.
Consumers Right to Choose
The International Federation of Organic Agriculture Movement has made stringent efforts to keep GE crops out of organic production, yet some US organic farmers have found their corn (maize) crops, including seeds, contain detectable levels of genetically engineered DNA. The organic movement is firm in its opposition to any use of GE crops in agriculture, and organic standards explicitly prohibit their use (however, keep in mind that even “organic” maize has incurred significant genetic modification compared to its wild relative, teosinte). The farmers, whose seed is contaminated, have been under rigid organic certification, which assures that they did not use any kind of genetically modified materials on their farms.
Any trace of GE crops must have come from outside their production areas. While the exact origin is unclear at this time, it is likely that the contamination has been caused by pollen drift from GE crop fields in surrounding areas. However, the contamination may have also come from the seed supply. Seed producers, who intended to supply GE crop-free seed, have also been confronted with genetic contamination and cannot guarantee that their seed is 100% GE crop-free.
Long-Term Ecological Effects
An early study indicated the pollen from a particular type of genetically modified corn may be harmful to the caterpillars of monarch butterflies, This type of corn, known as Bt corn, is genetically modified to produce a bacterial protein that acts as an insecticide. This trait is favorable because it reduces the amount of insecticides used by farmers. Pollen from Bt corn can be harmful to caterpillars, but only at very high concentrations. These concentrations are seldom reached in nature and follow-up studies have found the effect of Bt corn to be negligible.
NASEM’s GE Crop Report documents that the validity of that initial monarch study was questioned by other scientists and this ultimately led to a large, multi-national study funded by the US and Canada. They found that the vast majority of Bt corn grown did not represent a risk to monarchs. However, one strain of Bt corn did, and it was consequently removed from the market.
The GE Crop Report also mentioned a separate threat to monarch: loss of milkweed, which is critical to the butterfly’s lifecycle. Some GE crops are engineered to resist the herbicide glyphosate. Farmers using these crops can spray their entire field with the herbicide, which kills milkweed but not their GE crop. This can lower the amount of milkweed growing within the habitat range of monarchs. The report concluded that more studies are needed to quantify the actual impact this may be having on monarch populations.
Human Health Risk
At least some of the genes used in GE crops may not have been used in the food supply before, so GM foods may pose a potential risk for human health, such as producing new allergens. But this is also true of crops generated by traditional breeding practices (because both produce genetic modifications and thus new traits).
Like other ‘controversial’ scientific issues, the scientific consensus on GE crops is quite clear: they are safe. The UN’s Food and Agriculture Organization has concluded that risks to human and animal health from the use of GMOs are negligible. NASEM’s GE Crop Report found “no substantiated evidence of a difference in risks to human health between current commercially available genetically engineered (GE) crops and conventionally bred crops, nor did it find conclusive cause-and-effect evidence of environmental problems from the GE crops.” The American Medical Association’s Council on Science and Public Health, in 2012, stated that “Bioengineered foods have been consumed for close to 20 years, and during that time, no overt consequences on human health have been reported and/or substantiated in the peer-reviewed literature.” Similar statements have been made by the US National Resource Council and the American Association for the Advancement of Science, which publishes the preeminent scholarly journal, Science .
The potential of GE crops to be allergenic is one of the potential adverse health effects and it should continue to be studied, especially because some scientific evidences indicates that animals fed GE crops have been harmed. NASEM’s GE Crop Report concluded that when developing new crops, it is the product that should be studied for potential health and environmental risks, not the process that achieved that product. What this means is, because both traditional breeding practices and modern genetic engineering produce new traits through genetic modification, they both present potential risks. Thus, for the safety of the environment and human health, both should be adequately studied.
Intellectual Property Rights
Intellectual property rights are one of the important factors in the current debate on GE crops. GE crops can be patented by Agri-businesses, which can lead to them controlling and potentially exploiting agricultural markets. Some accuse companies, such as Monsanto, of allegedly controlling seed production and pricing, much to the detriment of farmers. NASEM’s GE Crop Report recommends more research into how the concentration of seed-markets by a few companies, and the subsequent reduction of free market competition, may be affecting seed prices and farmers.
It should be noted that crops developed by traditional breeding can also be legally protected and controlled in ways similar to GE crops. Jim Myers, from Oregon State University notes that “In all but a few cases, all contemporary varieties developed by private breeders are [legally] protected, and most public varieties are protected as well.”
Are GE Crops the Solution We Need?
Significant resources, both financial and intellectual, have been allocated to answering the question: are GE crops safe? After many hundreds of scientific studies, the answer is yes. But a significant question still remains: are they necessary? Certainly, such as in instances like Hawaii’s papaya, which were threated with eradication due to an aggressive disease, genetic engineering was a quick and effective solution that would have been extremely difficult, if not impossible, to solve using traditional breeding practices.
However, in many cases, the early promises of GE crops – that they would improve nutritional quality of foods, confer disease resistance, and provide unparalleled advances in crop yields – have largely failed to come to fruition. NASEM’s GE Crop Report states that while GE crops have resulted in the reduction of agricultural loss from pests, reduced pesticide use, and reduced rates of injury from insecticides for farm workers, they have not increased the rate at which crop yields are advancing when compared to non-GE crops. Additionally, while there are some notable exceptions like golden rice or virus-resistant papayas, very few GE crops have been produced to increase nutritional capacity or to prevent plant disease that can devastate a farmer’s income and reduce food security. The vast majority of GE crops are developed for only two purposes: to introduce herbicide resistance or pest resistance.
Genetic engineering of crops represents in important tool in a world of rapidly changing climate and a burgeoning human population, but as you will see in the next chapter, it is only one of many tools that agriculturists can use to produce enough food for all humans while simultaneously working to conserve the environment.
Suggested Supplementary Reading:
Achenbach, J. 2015. “Why Do Many Reasonable People Doubt Science?” National Geographic Magazine . https://www.nationalgeographic.com/magazine/2015/03/science-doubters-climate-change-vaccinations-gmos/
NASEM. 2016. Genetically Engineered Crops: Experiences and Prospects . http://nas-sites.org/ge-crops/category/report/
Contributors and Attributions
- The History of Genetic Modification of Crops and Are GE Crops the Solution We Need? by Matthew R. Fisher are licensed under CC BY 4.0 .
- Essentials of Environmental Science by Kamala Doršner is licensed under CC BY 4.0 . Modified from the original by Matthew R. Fisher.
- “Genetically Modified Foods and Social Concerns” by Behrokh Mohajer Maghari and Ali M. Ardekani is licensed under CC BY-NC 4.0 . Modified from the original by Matthew R. Fisher.


2.4: Inverse Functions
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A fashion designer is travelling to Milan for a fashion show. He asks his assistant, Betty, what 75 degrees Fahrenheit is in Celsius, and after a quick search on Google, she finds the formula \(C=\dfrac{5}{9} (F-32)\). Using this formula, she calculates \(\dfrac{5}{9} (75-32)\approx 24\)degrees Celsius. The next day, the designer sends his assistant the week’s weather forecast for Milan, and asks her to convert the temperatures to Fahrenheit.
At first, Betty might consider using the formula she has already found to do the conversions. After all, she knows her algebra, and can easily solve the equation for \(F\) after substituting a value for \(C\). For example, to convert 26 degrees Celsius, she could write:
\[26 = \dfrac{5}{9}(F - 32)\nonumber \]
\[26 \cdot \dfrac{9}{5} = F - 32\nonumber \]
\[F = 26 \cdot \dfrac{9}{5} + 32 \approx 79\nonumber \]
After considering this option for a moment, she realizes that solving the equation for each of the temperatures would get awfully tedious, and realizes that since evaluation is easier than solving, it would be much more convenient to have a different formula, one which takes the Celsius temperature and outputs the Fahrenheit temperature. This is the idea of an inverse function, where the input becomes the output and the output becomes the input.
Definition: Inverse function
If \(f(a)=b\), then a function \(g(x)\) is an inverse of \(f\) if \(g(b)=a\).
The inverse of \(f(x)\) is typically notated \(f^{-1} (x)\), which is read “\(f\) inverse of \(x\)”, so equivalently, if \(f(a)=b\) then \(f^{-1} (b)=a\).
Important: The raised -1 used in the notation for inverse functions is simply a notation, and does not designate an exponent or power of -1.
Example \(\PageIndex{1}\)
If for a particular function, \(f(2)=4\), what do we know about the inverse?
Solution
The inverse function reverses which quantity is input and which quantity is output, so if \(f(2)=4\), then \(f^{-1} (4)=2\).
Alternatively, if you want to re-name the inverse function \(g(x)\), then \(g(4) = 2\)
Exercise \(\PageIndex{1}\)
Given that \(h^{-1} (6)=2\), what do we know about the original function \(h(x)\)?
- Answer
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\(g(2) = 6\)
Notice that original function and the inverse function undo each other. If \(f(a)=b\), then \(f^{-1} (b)=a\), returning us to the original input. More simply put, if you compose these functions together you get the original input as your answer.
\[f^{-1} (f(a))=a\text{ and }f(f^{-1} (b))=b\]
Since the outputs of the function \(f\) are the inputs to \(f^{-1}\), the range of \(f\) is also the domain of \(f^{-1}\). Likewise, since the inputs to \(f\) are the outputs of \(f^{-1}\), the domain of \(f\) is the range of \(f^{-1}\).
Basically, like how the input and output values switch, the domain & ranges switch as well. But be careful, because sometimes a function doesn’t even have an inverse function, or only has an inverse on a limited domain. For example, the inverse of \(f(x)=\sqrt{x}\) is \(f^{-1} (x)=x^{2}\), since a square “undoes” a square root, but it is only the inverse of \(f(x)\) on the domain \([0, \infty)\), since that is the range of \(f(x)=\sqrt{x}\).
Example \(\PageIndex{2}\)
The function \(f(x)=2^{x}\) has domain \((-\infty ,\infty )\) and range \((0,\infty )\), what would we expect the domain and range of \(f^{-1}\) to be?
Solution
We would expect \(f^{-1}\) to swap the domain and range of \(f\), so \(f^{-1}\) would have domain \((0,\infty )\) and range \((-\infty ,\infty )\).
Example \(\PageIndex{3}\)
A function \(f(t)\) is given as a table below, showing distance in miles that a car has traveled in \(t\) minutes. Find and interpret \(f^{-1} (70)\)
| \(t\) (minutes) | 30 | 50 | 70 | 90 |
| \(f(t)\) (miles) | 20 | 40 | 60 | 70 |
Solution
The inverse function takes an output of \(f\) and returns an input for \(f\). So in the expression \(f^{-1} (70)\), the 70 is an output value of the original function, representing 70 miles. The inverse will return the corresponding input of the original function \(f\), 90 minutes, so \(f^{-1} (70)=90\). Interpreting this, it means that to drive 70 miles, it took 90 minutes.
Alternatively, recall the definition of the inverse was that if \(f(a)=b\) then \(f^{-1} (b)=a\). By this definition, if you are given \(f^{-1} (70)=a\) then you are looking for a value a so that \(f(a)=70\). In this case, we are looking for a \(t\) so that \(f(t)=70\), which is when \(t = 90\).
Exercise \(\PageIndex{2}\)
Using the table below
| \(t\) (minutes) | 30 | 50 | 60 | 70 | 90 |
| \(f(t)\) (miles) | 20 | 40 | 50 | 60 | 70 |
Find and interpret the following
a. \(f(60)\)
b. \(f^{-1} (60)\)
- Answer
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a. \(f(60) = 50\). In 60 minutes, 50 miles are traveled.
b. \(f^{-1}(60) = 70\). To travel 60 miles, it will take 70 minutes.
Example \(\PageIndex{4}\)
A function \(g(x)\) is given as a graph below. Find \(g(3)\) and \(g^{-1} (3)\)
Solution
To evaluate \(g(3)\), we find 3 on the horizontal axis and find the corresponding output value on the vertical axis. The point (3, 1) tells us that \(g(3)=1\)
To evaluate \(g^{-1} (3)\), recall that by definition \(g^{-1} (3)\) means \(g(x) = 3\). By looking for the output value 3 on the vertical axis we find the point (5, 3) on the graph, which means \(g(5) = 3\), so by definition \(g^{-1} (3) =5\).
Exercise \(\PageIndex{3}\)
Using the graph in Example 4 above
a. find \(g^{-1} (1)\)
b. estimate \(g^{-1} (4)\)
- Answer
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a. \(g^{-1} (1) = 3\)
b. \(g^{-1} (4) = 5.5\) (this is an approximation - answers may vary slightly)
Example \(\PageIndex{5}\)
Returning to our designer’s assistant, find a formula for the inverse function that gives Fahrenheit temperature given a Celsius temperature.
Solution
A quick Google search would find the inverse function, but alternatively, Betty might look back at how she solved for the Fahrenheit temperature for a specific Celsius value, and repeat the process in general
\[\begin{array}{l} {C=\dfrac{5}{9} (F-32)} \\ {C\cdot \dfrac{9}{5} =F-32} \\ {F=\dfrac{9}{5} C+32} \end{array}\nonumber \]
By solving in general, we have uncovered the inverse function. If
\[C=h(F)=\dfrac{5}{9} (F-32)\nonumber \]
Then
\[F=h^{-1} (C)=\dfrac{9}{5} C+32\nonumber \]
In this case, we introduced a function h to represent the conversion since the input and output variables are descriptive, and writing \(C^{-1}\)could get confusing.
It is important to note that not all functions will have an inverse function. Since the inverse \(f^{-1} (x)\) takes an output of \(f\) and returns an input of \(f\), in order for \(f^{-1}\) to itself be a function, then each output of \(f\) (input to \(f^{-1}\)) must correspond to exactly one input of \(f\) (output of \(f^{-1}\)) in order for \(f^{-1}\) to be a function. You might recall that this is the definition of a one-to-one function.
Definition: Properties of inverses
In order for a function to have an inverse , it must be a one-to-one function.
In some cases, it is desirable to have an inverse for a function even though the function is not one-to-one. In those cases, we can often limit the domain of the original function to an interval on which the function is one-to-one, then find an inverse only on that interval.
If you have not already done so, go back to the toolkit functions that were not one-to-one and limit or restrict the domain of the original function so that it is one-to-one. If you are not sure how to do this, proceed to Example 6.
Example \(\PageIndex{6}\)
The quadratic function \(h(x)=x^{2}\) is not one-to-one. Find a domain on which this function is one-to-one, and find the inverse on that domain.
Solution
We can limit the domain to \([0,\infty )\)to restrict the graph to a portion that is one-to-one, and find an inverse on this limited domain.
You may have already guessed that since we undo a square with a square root, the inverse of \(h(x)=x^{2}\) on this domain is \(h^{-1} (x)=\sqrt{x}\).
You can also solve for the inverse function algebraically. If \(h(x)=x^{2}\), we can introduce the variable \(y\) to represent the output values, allowing us to write \(y=x^{2}\). To find the inverse we solve for the input variable
To solve for \(x\) we take the square root of each side. \(\sqrt{y} =\sqrt{x^{2} }\) and get \(\sqrt{y} =|x|\), so \(x=\pm \sqrt{y}\). We have restricted \(x\) to being non-negative, so we’ll use the positive square root, \(x=\sqrt{y}\) or \(h^{-1} (y)=\sqrt{y}\). In cases like this where the variables are not descriptive, it is common to see the inverse function rewritten with the variable \(x\): \(h^{-1} (x)=\sqrt{x}\). Rewriting the inverse using the variable \(x\) is often required for graphing inverse functions using calculators or computers.
Note that the domain and range of the square root function do correspond with the range and domain of the quadratic function on the limited domain. In fact, if we graph \(h(x)\) on the restricted domain and \(h^{-1} (x)\) on the same axes, we can notice symmetry: the graph of \(h^{-1} (x)\) is the graph of h(x) reflected over the line \(y = x\).
Example \(\PageIndex{7}\)
Given the graph of \(f(x)\) shown, sketch a graph of \(f^{-1} (x)\).
Solution
This is a one-to-one function, so we will be able to sketch an inverse. Note that the graph shown has an apparent domain of \((0, \infty)\) and range of \((-\infty, \infty)\), so the inverse will have a domain of \((-\infty, \infty)\) and range of \((0, \infty)\).
Reflecting this graph of the line \(y = x\), the point (1, 0) reflects to (0, 1), and the point (4, 2) reflects to (2, 4). Sketching the inverse on the same axes as the original graph:
Important Topics of this Section
Definition of an inverse function
Composition of inverse functions yield the original input value
Not every function has an inverse function
To have an inverse a function must be one-to-one
Restricting the domain of functions that are not one-to-one.


11.8: Poeme Electronique
Here is a brief article on our playlist example of electronic music.
Introduction
Poème électronique (English Translation: “Electronic Poem”) is an 8-minute piece of electronic music by composer Edgard Varèse, written for the Philips Pavilion at the 1958 Brussels World’s Fair. The Philips corporation commissioned Le Corbusier to design the pavilion, which was intended as a showcase of their engineering progress. Le Corbusier came up with the title Poème électronique , saying he wanted to create a “poem in a bottle.” Varèse composed the piece with the intention of creating a liberation between sounds and as a result uses noises not usually considered “musical” throughout the piece.
 The first image is a bull’s head in a spotlight. The final image is a woman holding an infant. Le Corbusier assigned thematic sections to the film:
| 0 – 60″ | Genesis |
| 61 – 120″ | Spirit and Matter |
| 121 – 204″ | From Darkness to Dawn |
| 205 – 240″ | Man-Made Gods |
| 241 – 300″ | How Time Moulds Civilization |
| 301 – 360″ | Harmony |
| 361 – 480″ | To All Mankind |
The sequence of sounds in Varèse’s composition:
| 0″ | 1. | a. | Low bell tolls. “Wood blocks.” Sirens. Fast taps lead to high, piercing sounds. 2-second pause. |
| 43″ | b. | “Bongo” tones and higher grating noises. Sirens. Short “squawks.” Three-tone group stated three times. | |
| 1’11” | c. | Low sustained tones with grating noises. Sirens. Short “squawks.” Three-tone group. 2-second pause. | |
| 1’40” | d. | Short “squawks.” High “chirps.” Variety of “shots,” “honks,” “machine noises.” Sirens. Taps lead to | |
| 2’36” | 2. | a. | Low bell tolls. Sustained electronic tones. Repeated “bongo” tones. High and sustained electronic tones. Low tone, crescendo. Rhythmic noises lead to |
| 3’41” | b. | Voice, “Oh-gah.” 4-second pause. Voice continues softly. | |
| 4’17” | c. | Suddenly loud. Rhythmic percussive sounds joined by voice. Low “animal noises,” scraping, shuffling, hollow vocal sounds. Decrescendo into 7-second pause. | |
| 5’47” | d. | Sustained electronic tones, crescendo and decrecendo. Rhythmic percussive sounds. Higher sustained electronic tones, crescendo. “Airplane rumble,” “chimes,” jangling. | |
| 6’47” | e. | Female voice. Male chorus. Electronic noises, organ. High taps. Swooping organ sound. Three-note group stated twice. Rumble, sirens, crescendo (8 minutes and 5 seconds).” |
Contributors and Attributions
- Authored by : Elliott Jones. Provided by : Santa Ana College. Located at : http://www.sac.edu . License : CC BY: Attribution
- Poeme electronique. Provided by : Wikipedia. Located at : http://en.Wikipedia.org/wiki/Po%C3%A8me_%C3%A9lectronique . License : CC BY-SA: Attribution-ShareAlike


4.1: Course Project - Create a Social Media Strategy
Course Project: Social Media Strategy
Throughout this semester, you’ll develop a social media marketing strategy for a specific company or department within the university. You’ll pick a client/company (real or fictional), focusing on an organization with a decision-maker for whom you’ll design the marketing strategy. No need for actual access to the organization – ‘client’ simply means you can’t choose an industry for this project.
Due Dates:
Steps 1-2: February 23
Steps 3-4: March 9
Steps 5-7: April 6
Steps 8-10: May 4
We’ll also have assignments to create content for posting along the way. Your final submission will include a written proposal of all 10 steps and a short video (20 minutes max) to describe your strategy to the client. Due May 4 (In-person classes will present in class).
Step 1. Business Scope
Gather intel on the client’s mission, product/service, market, and competitors. If possible, define their Why, How, and What; if not, suggest creating one.
Step 2. Aiming High
Outline the client’s business and social media goals – engagement, growth, or ROI.
Step 3. Eye on the Customer
Identify target audiences using client criteria or platform data, and create customer personas with demographics and psychographics.
Step 4. Know the Competition
Examine competitors’ social media performance and compare it to the client’s. Identify standout tactics for both.
Step 5. Social Media Checkup
Assess what’s working or not, audience engagement, target audience network usage, and comparison to the competition.
Step 6. Tone It Up
Decide on a communication style for your strategy based on audience, personas, and platform suggestions. Include a hashtag strategy.
Step 7. Content Countdown
Pick the networks to use based on target audience and persona data. Create a content calendar for March, April, and May.
Step 8. Content Crafting
Develop content, captions, posting dates/times, including photos, videos, links, reposts, and captions.
Step 9. Platform Play
Choose platforms and software for scheduling, posting, and engaging with content (Meta Business Manager/ Creator Studio, Tailwind, Iconosquare, Buffer, Hootsuite, Creator Studio, IFTTT, etc.).
Step 10. Measure and Report
Submit your overall strategy, content plan, and any analytics available from posted content.
Turn in your written strategy and record a video describing your plan and any available analytics (for in-person class, present to the class).


6.2: Elliptic Geometry
As was the case in hyperbolic geometry, the space in elliptic geometry is derived from \(\mathbb{C}^+\text{,}\) and the group of transformations consists of certain Möbius transformations. We first consider the transformations.
Definition
Let \({\cal S}\) consist of all Möbius transformations \(T\) that preserve antipodal points. In other words, \(\cal S\) consists of all Möbius transformations \(T\) with the property that if \(z\) and \(z_a\) are antipodal points in \(\mathbb{C}^+\) then \(T(z)\) and \(T(z_a)\) are antipodal points in \(\mathbb{C}^+\text{.}\)
We leave it to the reader to verify that \(\cal{S}\) actually forms a group. Considering the antipodal point construction in Figure \(6.1.3\), it seems clear that rotation of \(\mathbb{C}^+\) about the origin should preserve antipodal points. So, these rotations should be in \(\cal S\text{.}\) Let's run through this.
Example \(6.2.1\): Rotations about \(0\) are in \({\cal S}\).
Suppose \(R(z)=e^{i\theta}z\) is a rotation about the origin and \(z\) and \(z_a\) are antipodal points. If \(z = \infty\) or \(0\text{,}\) then rotation about the origin fixes \(z\text{,}\) and it also fixes its antipodal point, so the map \(R\) preserves antipodal points in these cases. Now suppose \(z \neq 0, \infty\text{.}\) Then \(z_a = -\dfrac{1}{\overline{z}}\text{.}\) Since \(\overline{R(z_a)} = -\dfrac{e^{-i\theta}}{z}\text{,}\) it follows that
\[ R(z)\cdot \overline{R(z_a)} = -e^{i\theta}z \cdot \dfrac{e^{-i\theta}}{z} = -1\text{.} \]
Thus, the image points \(R(z)\) and \(R(z_a)\) are still antipodal points. Rotations about the origin belong to the group \(\cal S\text{.}\)
Now we consider what a typical transformation in \(\cal S\) looks like.
Suppose the Möbius transformation \(T\) is in \(\cal S\text{,}\) and that \(z\) and \(w\) are antipodal points. Then \(T(z)\cdot \overline{T(w)} = -1\text{.}\) Since \(z\) and \(w\) are antipodal points, \(w = -\dfrac{1}{\overline{z}}\text{,}\) so \(T(z)\cdot \overline{T(-1/\bar z)} = -1\text{,}\) or
\[\begin{align} T(z) & = \dfrac{-1}{\overline{T(\dfrac{-1}{\bar z})}}\text{.} \end{align}\]
Assume \(T(z) = \dfrac{(az + b)}{(cz + d)}\text{,}\) and that it has determinant \(ad- bc = 1\text{.}\) (Recall, from Exercise \(3.4.4\) that we can always write a Möbius transformation in a form with determinant \(1\), and this form is unique up to sign.) Now,
\begin{align*} \overline{T(-\dfrac{1}{\bar z})} & = \overline{\left[\dfrac{\frac{-a}{\bar z + b}}{\frac{-c}{\bar z + d}}\right]}\\ & =\dfrac{-\bar{a}/{z} + \bar{b}}{-\bar{c}/z +\bar{d}}\\ & =\dfrac{\bar b z - \bar a}{\bar{d} z - \bar{c}}\text{.} \end{align*}
Substituting into equation \(6.2.2\) of this derivation yields
\[ \dfrac{az+b}{cz + d} = \dfrac{-\bar d z + \bar c}{\bar b z - \bar a}\text{.} \]
The transformation on the right also has determinant one, so these two transformations are identical up to sign. We can assume that \(d = -\bar a\) and that \(c = \bar b\text{,}\) so that \(T\) may be expressed as follows:
\[ T(z) = \dfrac{az + b}{\bar b z - \bar a}\text{.} \]
We can make this general form look a lot like the general form for a transformation in \({\cal H}\text{.}\) To do so, first multiply each term by \(-\dfrac{1}{\bar a}\text{,}\) assuming \(a \neq 0\text{.}\) (If \(a = 0\text{,}\) what would the transformation look like?)
\begin{align*} T(z) & = \dfrac{-\frac{a}{\bar a} z - \dfrac{b}{\bar a}}{-\dfrac{\bar b}{\bar a} z + 1}\\ & = \dfrac{-\frac{a}{\bar a}(z + \dfrac{b}{a})}{-\frac{\bar b}{\bar a} z + 1}\\ & = e^{i\theta}\cdot \dfrac{z-z_0}{\overline{z}_0z + 1}\text{,} \end{align*}
where \(e^{i\theta} = -\dfrac{a}{\bar{a}}\) and \(z_0 = -\dfrac{b}{a}\text{.}\) Thus, we have derived the following algebraic description of transformations in \(\cal S\text{.}\)
Any transformation \(T\) in the group \(\cal S\) has the form
\[ T(z) = e^{i\theta}\frac{z - z_0}{1+\overline{z}_0z} \]
for some angle \(\theta\) and some \(z_0\) in \(\mathbb{C}^+\text{.}\)
One can show that the following converse holds: Any transformation having the form above preserves antipodal points. It follows that for any point \(z_0 \in \mathbb{C}^+\text{,}\) there exists a transformation \(T\) in \(\cal S\) such that \(T(z_0) = 0\text{,}\) and since rotations about the origin live in \(\cal S\) we can prove the following useful result (see Exercise \(6.2.1\)).
Given distinct points \(z_0\) and \(z_1\) in \(\mathbb{C}^+\text{,}\) there exists a transformation in \({\cal S}\) that sends \(z_0\) to 0 and \(z_1\) to the real axis.
We now prove a theorem that helps us visualize the maps in \({\cal S}\text{.}\)
If a Möbius transformation preserves antipodal points, then it is an elliptic Möbius transformation.
- Proof
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Suppose \(T\) is a Möbius transformation that preserves antipodal points. If \(T\) is not the identity map then it must fix one or two points. However, if \(T\) preserves antipodal points and fixes a point \(p\text{,}\) then it must fix its antipodal point \(p_a\text{.}\) Thus, \(T\) must have two fixed points and \(T\) has normal form
\[ \frac{T(z)-p}{T(z)-p_a} = \lambda\frac{z-p}{z-p_a}\text{.} \]
To show \(T\) is an elliptic Möbius transformation, we must show that \(|\lambda| = 1\text{.}\) Setting \(z = 0\) the normal form reduces to
\[ \frac{T(0) - p}{T(0)-p_a} = \lambda\frac{p}{p_a}\text{.} \]
Solve for \(T(0)\) to obtain
\[ T(0) = \frac{pp_a - \lambda pp_a}{p_a - \lambda p}\text{.} \]
On the other hand, setting \(z = \infty\) and solving for \(T(\infty)\text{,}\) one checks that
\[ T(\infty) =\frac{p-\lambda p_a}{1-\lambda}\text{.} \]
Since \(T\) preserves antipodal points, \(T(0)\cdot \overline{T(\infty)} = -1\text{;}\) therefore, we have
\[ \frac{pp_a - \lambda pp_a}{p_a - \lambda p} \cdot \frac{\overline{p}-\overline{\lambda}\overline{p_a}}{1-\overline{\lambda}} = -1\text{.} \]
If we expand this expression and solve it for \(\lambda\overline{\lambda}\) (using the fact that \(p\cdot \overline{p_a} = -1\text{,}\) and \(p \neq p_a\)), we obtain \(\lambda \overline{\lambda} = 1\text{,}\) from which it follows that \(|\lambda| = 1\text{.}\) Thus \(T\) is an elliptic Möbius transformation.
In Exercise \(3.5.8\) we showed that any elliptic Möbius transformation is the composition of two inversions about clines that intersect at the two fixed points. In the case of a Möbius transformation that preserves antipodal points, these two fixed points must be antipodal to each other. It follows by Lemma \(6.1.2\) that the two clines of inversion are great circles in \(\mathbb{C}^+\text{.}\) Thus, we may view each transformation of \(\cal S\) as the composition of two inversions about great circles. This is reassuring. By Theorem \(6.1.1\), these inversions correspond to reflections of the sphere about great circles, and composing two of these reflections of the sphere yields a rotation of the sphere. The transformation group \(\cal S\text{,}\) then, consists of Möbius transformations that correspond via stereographic projection to rotations of the sphere. We summarize these facts below.
Any transformation in \(\cal S\) is the composition of two inversions about great circles in \(\mathbb{C}^+\text{,}\) and it corresponds via stereographic projection to a rotation of the unit \(2\)-sphere.
Before turning to the space in elliptic geometry, we make one more comment about the group \({\cal S}\text{.}\) The reader may have noticed a strong similarity, algebraically, between the maps in \({\cal S}\) and the maps in \({\cal H}\text{,}\) the transformation group in hyperbolic geometry. Recall, transformations in \({\cal H}\) have the form
\begin{align*} T(z) & = e^{i\theta}\frac{z - z_0}{1-\overline{z}_0z}\tag{transformation in $\cal{H}$}\text{.} \end{align*}
The single sign difference between the algebraic desciption of maps in \(\cal S\) from the maps in \(\cal H\) is not a coincidence. We could have defined the transformations in hyperbolic geometry as those Möbius transformations that preserve symmetric points with respect to the unit circle. That is, we could have defined the group \(\cal{H}\) to consist of all Möbius transformations satisfying this property: If \(z \cdot \bar w = 1\) then \(T(z) \cdot \overline{T(w)} = 1\text{.}\) If we had then asked what such a \(T\) would look like, we would have gone through the argument as we did in this section, with a \(+1\) initially instead of a -1. This sign difference propagates to the sign difference in the final form of the description of the map.
Subsection t he Space for Elliptic Geometry
One could let the space be all of \(\mathbb{C}^+\text{.}\) If we take this tack, then we are reproducing the geometry of the sphere in \(\mathbb{C}^+\text{.}\) We call the geometry \((\mathbb{C}^+,{\cal S})\) spherical geometry . Distance can be defined to match distance on the unit sphere, and great cirlces in \(\mathbb{C}^+\) will be geodesics. Rather than develop these details with this choice of spaces, we will focus instead on a different space. We do so because we want to build a geometry in which there is a unique line between any two points. This is not quite true on the sphere, and so not quite true in \((\mathbb{C}^+,{\cal S})\text{.}\) If two points in \(\mathbb{C}^+\) are antipodal points, such as 0 and \(\infty\text{,}\) then there are infinitely many lines (great circles) through these points.
So, we choose to work primarily with a space in which this feature of many distinct lines through two points vanishes. The trick is to identify antipodal points . That is, the space we will consider is actually the space \(\mathbb{C}^+\) with antipodal points identified. What does this space look like?
Remember the flat torus from Chapter 1? Each point on the boundary of the rectangle is identified with the corresponding point on the opposite edge. The two points are fused together into a single point.
In elliptic space, every point gets fused together with another point, its antipodal point. So, for instance, the point \(2 + i\) gets identified with its antipodal point \(-\dfrac{2}{5}-\dfrac{i}{5}\text{.}\) In elliptic space, these points are one and the same.
With the flat torus, we could visualize the space after identifying points by wrapping it up in three-dimensional space. This was possible since we were only identifying the edges of the rectangle. In the present case every point in \(\mathbb{C}^+\) gets paired up. To help visualize the space here we look at a region in the plane that contains one representative from each pair.
Consider the closed unit disk, consisting of all complex numbers \(z\) such that \(|z| \leq 1\text{.}\) For each point \(w\) outside this disk, its antipodal point \(w_a\) is inside the disk. Thus, the closed unit disk contains a representative from each pair of antipodal points. However, there is some redundancy: for a point \(w\) on the boundary of the disk \((|w| = 1)\text{,}\) its antipodal point \(w_a\) is also on the boundary. To account for this redundancy, we identify each point on the boundary of the closed unit disk with its antipodal point.
This will be our model for the space in elliptic geometry, and this space is called the projective plane.
The projective plane , denoted \(\mathbb{P}^2\text{,}\) consists of all complex numbers \(z\) such that \(|z| \leq 1\) with the additional feature that antipodal points on the unit circle are identified.
We can think of this space as the closed unit disk with its two edges (top-half circle and bottom-half circle) identified according to the arrows in Figure \(6.2.1\). Notice the pleasant journey a bug has taken from \(p\) to \(q\) in this figure. From \(p\) she heads off toward point \(c\text{,}\) which appears on the boundary of our model. When she arrives there she simply keeps walking, though in our model we see her leave the “screen” and reappear at the antipodal point \(c_a\text{.}\) She has her sights set on point \(d\) and saunters down there, continues on (reappearing at \(d_a\)), and heads on to \(q\text{,}\) hungry but content.
The disk model for elliptic geometry , \((\mathbb{P}^2,{\cal S})\text{,}\) is the geometry whose space is \(\mathbb{P}^2\) and whose group of transformations \(\cal{S}\) consists of all Möbius transformations that preserve antipodal points.
Because the transformations of \(\cal S\) are generated by inversions about great circles, these circles ought to determine the lines in elliptic geometry.
An elliptic line in \((\mathbb{P}^2,{\cal S})\) is the portion of a great circle in \(\mathbb{C}^+\) that lives in the closed unit disk.
Two elliptic lines have been constructed in Figure \(6.2.2\).
There is a unique elliptic line connecting two points \(p\) and \(q\) in \(\mathbb{P}^2\text{.}\)
- Proof
-
Suppose \(p\) and \(q\) are distinct points in \(\mathbb{P}^2\text{.}\) This means \(q \neq p_a\) as points in \(\mathbb{C}^+\text{.}\) Construct the antipodal point \(p_a\text{,}\) which gives us three distinct points in \(\mathbb{C}^+\text{:}\) \(p,q\) and \(p_a\text{.}\) There exists a unique cline through these three points. Since this cline goes through \(p\) and \(p_a\text{,}\) it is an elliptic line by Lemma \(6.1.2\).
Note that elliptic lines through the origin are Euclidean lines, just as was the case in the Poincaré model of hyperbolic geometry. As a result, to prove facts about elliptic geometry, it can be convenient to transform a general picture to the special case where the origin is involved.
The set of elliptic lines is a minimally invariant set of elliptic geometry.
- Proof
-
By definition, any transformation \(T\) in \(\cal{S}\) preserves antipodal points. Thus, if \(L\) is an elliptic line, then \(T(L)\) is as well, and the set of elliptic lines is an invariant set of elliptic geometry.
To show the set is minimally invariant, we appeal to Theorem \(4.1.1\), and prove that any two elliptic lines are congruent. To see this, notice that any elliptic line \(L\) is congruent to the elliptic line on the real axis. Indeed, for any points \(z_0\) and \(z_1\) on \(L\text{,}\) Lemma 6.2.3 ensures the existence of a transformation \(T\) in \(\cal{S}\) that sends \(z_0\) to the origin, and \(z_1\) to the real axis. It follows that \(T(L)\) is the real axis. Since all elliptic lines are congruent to the real axis, any two elliptic lines are congruent.
Any two elliptic lines intersect in \(\mathbb{P}^2\text{.}\)
- Proof
-
Given any two elliptic lines, apply a transformation \(T\) in \(\cal{S}\) that sends one of them to the real axis. It is enough to prove that any elliptic line in \(\mathbb{P}^2\) must intersect the real axis. Suppose \(M\) is an arbitrary elliptic line in \(\mathbb{P}^2\) and \(z\) is a point on \(M\text{.}\) If Im\((z) = 0\) then \(z\) is on the real axis and we are done.
If Im\((z) > 0\text{,}\) then \(z\) lies above the real axis. It follows that Im\((z_a)\lt 0\) by the definition of the antipodal point \(z_a\text{.}\) Since \(M\) contains both \(z\) and \(z_a\) it intersects the real axis at some point.
If Im\((z) \lt 0\text{,}\) then Im\((z_a) > 0\) and \(M\) must intersect the real axis as before. In either case, \(M\) must intersect the real axis, and it follows that any two elliptic lines must intersect.
As an immediate consequence of this theorem, there is no notion of parallel lines in elliptic geometry.
If \(p\) in \(\mathbb{P}^2\) is not on the elliptic line \(L\text{,}\) then every elliptic line through \(p\) intersects \(L\text{.}\)
Exercises
The transformation group in elliptic geometry.
- Prove that \(\cal{S}\) is a group of transformations.
-
For each \(\theta \in \mathbb{R}\) and \(z_0 \in \mathbb{C}\text{,}\) prove that the following Möbius transformation is in \(\cal S\text{:}\)
\[ T(z)=e^{i\theta}\dfrac{z-z_0}{1+\overline{z_0}z}\text{.} \]
- For each \(\theta \in \mathbb{R}\text{,}\) prove that \(T(z) = e^{i\theta}\frac{1}{z}\) is in \(\cal S\text{.}\)
- Use (b) and (c) to prove that for any distinct points \(p, q \in \mathbb{C}^+\) there exists a transformation in \(\cal S\) that sends \(p\) to \(0\) and \(q\) to a point on the positive real axis, thus proving Lemma \(6.2.1\).
Prove that the disk model for elliptic geometry is homogeneous.
Given a point \(z\) not on an elliptic line \(L\text{,}\) prove there exists an elliptic line through \(z\) that intersects \(L\) at right angles.
- Hint
-
First transform \(z\) to a convenient location in \(\mathbb{P}^2\text{.}\)
Is there a nonidentity transformation in \(\cal{S}\) that fixes two distinct points in \(\mathbb{P}^2\text{?}\) If so, find one, otherwise explain why no such transformation exists.
Find a transformation in \(\cal{S}\) that sends the point \(\dfrac{1}{2}\) to the point \(\dfrac{1}{2} + \dfrac{1}{2}i\text{.}\)