## Posts Tagged ‘**topology**’

## “It might help to think of the universe as a rubber sheet, or perhaps not”*…

You have most likely encountered one-sided objects hundreds of times in your daily life – like the universal symbol for recycling, found printed on the backs of aluminum cans and plastic bottles.

This mathematical object is called a Mobius strip. It has fascinated environmentalists, artists, engineers, mathematicians and many others ever since its discovery in 1858 by August Möbius, a German mathematician who died 150 years ago, on Sept. 26, 1868.

Möbius discovered the one-sided strip in 1858 while serving as the chair of astronomy and higher mechanics at the University of Leipzig. (Another mathematician named Listing actually described it a few months earlier, but did not publish his work until 1861.)…

The discovery of the Möbius strip in the mid-19th century launched a brand new field of mathematics: topology: “The Mathematical Madness of Möbius Strips and Other One-Sided Objects.”

*Hogfather*

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**As we return from whence we came,** we might wish a Joyeux Anniversaire to Denis Diderot, contributor to and the chief editor of the *Encyclopédie* (“All things must be examined, debated, investigated without exception and without regard for anyone’s feelings.”)– and thus towering figure in the Enlightenment; he was born on this date in 1713. Diderot was also a novelist (e.g., Jacques le fataliste et son maître [Jacques the Fatalist and his Master])… and no mean epigramist:

From fanaticism to barbarism is only one step.

We swallow greedily any lie that flatters us, but we sip only little by little at a truth we find bitter.

Man will never be free until the last king is strangled with the entrails of the last priest.

A thing is not proved just because no one has ever questioned it.

## “Woe, destruction, ruin, and decay; the worst is death and death will have his day”*…

Not long ago an old matchbook laying on photographer Pablo Iglesias Maurer‘s desk caught his eye. Or rather, it was the postcard-like picture on it, of a resort complex built in the 1960s. It got Pablo wondering how the place looked now, and the answer has led him to make an amazing photo series called

Abandoned States.The picture came with the title

How to Run A Successful Golf Course, but when Maurer got to the place, it was clear the owner of Penn Hills Resort didn’t follow that advice. He pointed the camera at the decaying building at roughly the same spot and did a ‘5-decades-after’ shot of the place.Ever since then, Pablo was hooked. He ordered more 60s postcards from eBay and started going around the country capturing these once beautiful buildings that now stand abandoned only as faint memories of what once was…

See more of his results at “Photographer Finds Locations Of 1960s Postcards To See How They Look Today, And The Difference Is Unbelievable” and here.

* Shakespeare, *Richard II*

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**As we contemplate continuity,** we might send never-ending birthday greetings to August Ferdinand Möbius; he was born on this date in 1790. A German mathematician and theoretical astronomer, he is best remembered as a topologist, more specifically for his discovery of the Möbius strip (a two-dimensional surface with only one side… or more precisely, a non-orientable two-dimensional surface with only one side when embedded in three-dimensional Euclidean space).

## “Sounds are quite innoxious, or most distressing, by their sort rather than their quantity”*…

More than 20 million people in the U.S. are afraid of flying. Sitting in a chair that’s floating in the air may be technologically stunning to some, but that floating-in-a-tin-can feeling puts some passengers on edge and sends their minds racing: Do the flight attendants look worried? What was that bump? And, oh man, what was that

noise?!But you don’t have to worry. You’re more likely to drown in your own bathtub than you are to perish in an out-of-control flight. In fact, the last time a U.S.-registered airliner had any fatalities was in 2009.

So unless the sound you hear is the flight attendants telling you to assume a bracing position—which really only means there’s the

potentialfor a problem—everything’s most likely O.K. Still, the unknown can be scary…

A breakdown—by sound—of most things you’ll hear on a flight and what each of those noises means: “A Nervous Flyer’s Guide to Every Ding, Buzz and Whir You Hear on an Airplane.”

* Jane Austen, *Persuasion*

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**As we assume the crash position,** we might send never-ending birthday greetings to August Ferdinand Möbius; he was born on this date in 1790. A German mathematician and theoretical astronomer, he is best remembered as a topologist, more specifically for his discovery of the Möbius strip (a two-dimensional surface with only one side… or more precisely, a non-orientable two-dimensional surface with only one side when embedded in three-dimensional Euclidean space).

## “Life is like topography”*…

*email readers click here for video*

What if you could see Earth’s 5-billion year journey not just in a book or on screen, but on the planet’s very topography? That’s the idea behind the audio-visual performance

Revolution of Topography, Cappadocia: Epic History of Humanity, which features 3D animations projection mapped onto the rocky surface of the Cappadocia Zelve Valley.Produced by FikirbazZenger and directed by Ferdi Alıcı,

Revolution of Topographyhas been billed as the world’s “largest mountain surface mapping”; and, with a 10-year screening time, will run for longer than any other projection mapping installation in history. The a/v installation, located at Cappadocia Zelve Valley Open Air Museum, will run through all phases of Cappadocia’s history, from geographical formation and topographical transformations to the emergence of civilization and religion…

More at “Mapping a Valley with Earth’s 5-Billion Year Journey.”

* “Life is like topography, Hobbes. There are summits of happiness and success, flat stretches of boring routine and valleys of frustration and failure.”

– “Calvin,” Bill Watterson

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**As we take the long view,** we might send spatially-sophisticated birthday greetings to William Paul Thurston; he was born on this date in 1946. A pioneer in the field of low-dimensional topology, he was awarded the 1982 Fields Medal for his contributions to the study of 3-manifolds. In later years, while his research continued, Thurston took on the challenge of mathematical popularization and education. He served as mathematics editor for *Quantum Magazine*, a youth science magazine, and was one of the founders of The Geometry Center. As director of Mathematical Sciences Research Institute from 1992 to 1997, he started a number of programs designed to increase awareness of mathematics among the public.

## “A child[’s]…first geometrical discoveries are topological…If you ask him to copy a square or a triangle, he draws a closed circle”*…

Topology is the Silly Putty of mathematics. Indeed, sometimes, topology is called “rubber-sheet geometry” because topologists study the properties of shapes that don’t change when an object is stretched or distorted. As Cliff Pickover explains, this leads to the creation of some pretty confounding shapes…

Mathematicians continue to invent strange objects to test their intuitions. Alexander’s horned sphere [above] is an example of a convoluted, intertwined surface for which it is difficult to define an inside and outside. Introduced by mathematician James Waddell Alexander (1888 – 1971), Alexander’s horned sphere is formed by successively growing pairs of horns that are almost interlocked and whose end points approach each other. The initial steps of the construction can be visualized with your fingers. Move the thumb and forefinger of each of your hands close to one another, then grow a smaller thumb and forefinger on each of these, and continue this budding without limit!

Although this may be hard to visualize, Alexander’s horned sphere is homeomorphic to a ball. In this case, this means that it can be stretched into a ball without puncturing or breaking it. Perhaps it is easier to visualize the reverse: stretching the ball into the horned sphere without ripping it. The boundary is, therefore, homeomorphic to a sphere…

Read more at “**The Official Alexander Sphere Appreciation Page**.”

* Jean Piaget

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**As we twist and turn,** we might send artfully-folded birthday greetings to Sir Erik Christopher Zeeman; he was born on this date in 1925. While he is probably most-widely known as a popularizer of Catastrophe Theory, his primary contributions to math have been in topology, more particularly in geometric topology (e.g., in knot theory) and in dynamical systems. The *Christopher Zeeman Medal for Communication of Mathematics* of the London Mathematical Society and the Institute of Mathematics and its Applications is named in his honor.

We might also spare a thought for Satyendra Nath Bose; he died on his date in 1974. A physicist and mathematician, he collaborated with Albert Einstein to develop a theory of statistical quantum mechanics, now called Bose-Einstein statistics. Paul Dirac named the class of particles that obey Bose–Einstein statistics, bosons, after Bose.

## “In a flat country a hillock thinks itself a mountain”*…

In 2003, the Annals of Improbable Research released the results of a study that was not so much groundbreaking as it was ground-battering: Kansas, the tongue-in-cheek analysis found, was flatter than a pancake. The researchers Mark Fonstad, William Pugatch, and Brandon Vogt used polynomial equations to calculate the flatness of the famously flat state, and discovered that—as compared to the topography of an IHOP pancake—it was indeed flatter than a flapjack.

Their finding was not incorrect. Parts of Kansas are, in fact, flatter than a pancake! But the study’s focus on Kanas, it turns out, was also misleading. Because there are states—six of them, to be specific—that are even flatter than Kansas. The states flatter than a pancake, you could say, could be served in a short stack.

This latest flatness finding comes courtesy of geographers at the University of Kansas, who just published a paper, “The Flatness of U.S. States,” in

Geographical Review, a peer-reviewed journal published by the American Geographical Society…The top 10 flattest states, per their results? [Results charted on the map above differ as they reflect a slightly different analysis; c.f., the link below.]

Florida

Illinois

North Dakota

Louisiana

Minnesota

Delaware

Kansas

Texas

Nevada

Indiana

Get level at “Science: Several U.S. States, Led by Florida, Are Flatter Than a Pancake.”

* Turkish proverb

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**As we reach for the maple syrup,** we might send lofty birthday greetings to Albert William Stevens; he was born on this date in 1886. An career officer in the U.S. Army Air Corps. Stevens was a pioneering balloonist and aerial photographer who took the first photograph clearly showing the Earth’s curvature (1930) and the first photographs of the Moon’s shadow on the Earth during a solar eclipse (1932). In 1935 Stevens and a colleague made a record balloon ascent near Rapid City, South Dakota. 20,000 watched– and millions listened to a live NBC broadcast– as their sealed gondola, Explorer II, climbed to 72,395 feet, nearly 14 miles, a record that stood until 1956.

## “The future ain’t what it used to be….”*

Just over a hundred years ago, in 1910, the *Cedar Rapids Evening Gazette* in Iowa published a list of advances and innovations that they believed would appear during the century, a fascinating list of things scientific/technical and social/political:

- Cure for cancer.
- Discovery south pole.
- Prevent or cure insanity.
- Influence sex by parental treatment.
- Create living organisms by artificial means.
- Phonograph records substitute for letter.
- Rationing clothing reform, health, comfort, durability only considerations
- Settle question of communication with Mars. Wonderful astronomical discoveries.
- Power of mind over matter a practical science devoid of superstitious elements.
- United States constitution rewritten, providing improved means for conservation of original democratic principles.
- Marvelous progress in transportation, largely aerial; airships and dirigible balloons crossing oceans and continents in remarkable time. Racing planes make five miles per minute. Inland waterways carry slow freight by improved methods. Monorail supplants two tracks. Electricity replaces steam. Convenient, economical city traffic system broadens city areas, opening suburban lands to householders. Pneumatic tubes for mails and express. Horses curiosities. Automobiles relegated to short distance burden bearing. Ocean steamers for freight, improvement toward speed rather than size.
- Produce rainfall at will.
- Temper gold and copper.
- Roads of nation paved.
- Conservation of sun’s heat and power.
- Cure for and elimination of tuberculosis.
- Development psychic research with fraud eliminated.
- Movements for universal language, universal religion, universal money.
- Non-existence of blindness by eliminating causes except accidents.
- Construction largely of concrete and metal or newly discovered materials.
- Electricity will move world’s wheels. Later radio-activity may substitute.
- Terrors of war so multiplied by death dealing inventions, chances of war minimized.
- Utilization of all energy, reducing consumption of wood and coal. Many fuel substitutes.
- Population of United States based on present ratio of increased, 1,317,547,000 at opening of twenty-first century.
- Rational diet with greatly reduced consumption of matter with increased nourishment from proper mastication and choices of foods.
- Machinery largely substituting manual energy, will promote pursuit of finer arts and sciences; give ample opportunity for relaxation and amusement; emancipate wage slaves. Three-hour work day predicted.
- Sea water for irrigation.
- Photographs in natural colors.
- Women’s political equality.
- Government control of corporations.
- Animated pictures in natural colors, transmitted by wireless.
- Substitution of heavier metals with aluminum, etc.
- Natural colors reproduced in newspaper pictures.
- Reduction of elimination all forms of gambling, including stocks.
- General acceptance of public ownership or control of public utilities.
- Government operation banking system, elimination of private banks. Postal savings banks.
- Moral, intellectual and economical awakening in dark sections of Africa, China a world power.
- Beautiful and healthful cities, offering with homes and work places all forms of free amusement, culture and recreation.
- Greater premium on brains with corresponding decreased in respect for position not gained by individual achievement.
- Revision judicial system, deciding causes on improved scientific plan, insuring equal justice. Pathological and psychological treatment for criminals. Crime reduced.
- Due to universal education, with special reference to hygiene, doctors and drugs be largely eliminated; average age to be near 60 years; men taller, stronger, higher intelligence and morals.

Some of their predictions were spot on (“Natural colors reproduced in newspaper pictures”); some, sadly as yet unrealized (“Cure for cancer”); and some, ironically backwards (“Government control of corporations”). But overall, it’s consoling to be reminded that things that seem wild, even radical at one moment in time– in this case, things like women’s rights and child labor laws– can, with the passage of time, become so obvious as to become human rights that we take for granted.

[TotH to Paleofuture, from whence the illustration above; via the always-excellent Next Draft]

* Yogi Berra

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**As we live in the past,** we might send twisted birthday greetings to August Ferdinand Möbius; he was born on this date in 1790. A mathematician and theoretical astronomer, Möbius was so important to the fields of analytic geometry, topology, and number theory that several mathematical concepts are named after him, including the Möbius configuration, the Möbius transformations, the Möbius transform, the Möbius function μ(*n*), and the Möbius inversion formula…

But he is best remembered, of course, as the creator of the Möbius Strip— a two-dimensional surface with only one side…. more specifically: a non-orientable two-dimensional surface with only one side when embedded in three-dimensional Euclidean space… It can be constructed in three dimensions: Take a rectangular strip of paper and join the two ends of the strip together so that it has a 180 degree twist. It is now possible to start at a point A on the surface and trace out a path that passes through the point which is apparently on the other side of the surface from A.