Posts Tagged ‘geometry’
“All you really need to know for the moment is that the universe is a lot more complicated than you might think, even if you start from a position of thinking it’s pretty damn complicated in the first place”*…
When you gaze out at the night sky, space seems to extend forever in all directions. That’s our mental model for the universe, but it’s not necessarily correct. There was a time, after all, when everyone thought the Earth was flat, because our planet’s curvature was too subtle to detect and a spherical Earth was unfathomable.
Today, we know the Earth is shaped like a sphere. But most of us give little thought to the shape of the universe. Just as the sphere offered an alternative to a flat Earth, other three-dimensional shapes offer alternatives to “ordinary” infinite space.
We can ask two separate but interrelated questions about the shape of the universe. One is about its geometry: the fine-grained local measurements of things like angles and areas. The other is about its topology: how these local pieces are stitched together into an overarching shape.
Cosmological evidence suggests that the part of the universe we can see is smooth and homogeneous, at least approximately. The local fabric of space looks much the same at every point and in every direction. Only three geometries fit this description: flat, spherical and hyperbolic…
Alternatives to “ordinary” infinite space: “What Is the Geometry of the Universe?”
* Douglas Adams, The Hitchhiker’s Guide to the Galaxy
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As we tinker with topology, we might recall that it was on this date in 1811 that Percy Bysshe Shelley was expelled from the University of Oxford for publishing the pamphlet The Necessity of Atheism. Shelley, of course, went on to become a celebrated lyric poet and one of the leaders of the English Romantic movement… one who had a confident (if not to say exalted) sense of his role in society:
Poets are the hierophants of an unapprehended inspiration; the mirrors of the gigantic shadows which futurity casts upon the present; the words which express what they understand not; the trumpets which sing to battle, and feel not what they inspire; the influence which is moved not, but moves. Poets are the unacknowledged legislators of the world.
“The laws of nature are but the mathematical thoughts of God”*…
2,300 years ago, Euclid of Alexandria sat with a reed pen–a humble, sliced stalk of grass–and wrote down the foundational laws that we’ve come to call geometry. Now his beautiful work is available for the first time as an interactive website.
Euclid’s Elements was first published in 300 B.C. as a compilation of the foundational geometrical proofs established by the ancient Greek. It became the world’s oldest, continuously used mathematical textbook. Then in 1847, mathematician Oliver Byrne rereleased the text with a new, watershed use of graphics. While Euclid’s version had basic sketches, Byrne reimagined the proofs in a modernist, graphic language based upon the three primary colors to keep it all straight. Byrne’s use of color made his book expensive to reproduce and therefore scarce, but Byrne’s edition has been recognized as an important piece of data visualization history all the same…
Explore elemental beauty at “A masterpiece of ancient data viz, reinvented as a gorgeous website.”
* Euclid, Elements
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As we appreciate the angles, we might spare a thought for Kurt Friedrich Gödel; he died on this date in 1978. A logician, mathematician, and philosopher, he is considered (along with Aristotle, Alfred Tarski— whose birthday this also is– and Gottlob Frege) to be one of the most important logicians in history. Gödel had an immense impact upon scientific and philosophical thinking in the 20th century. He is, perhaps, best remembered for his Incompleteness Theorems, which led to (among other important results) Alan Turing’s insights into computational theory.
Kurt Gödel’s achievement in modern logic is singular and monumental – indeed it is more than a monument, it is a landmark which will remain visible far in space and time. … The subject of logic has certainly completely changed its nature and possibilities with Gödel’s achievement. — John von Neumann
“The endless repetition of an ordinary miracle”*…
In 1611 Johannes Kepler wrote a scientific essay entitled De Nive Sexangula; commonly translated as “On the Six-Cornered Snowflake.” It was the first investigation into the nature of snowflakes and what we’d now call crystallography. Since he was a gentleman and a scholar back when you could be such a thing without being ironic or a hipster, Kepler gave the essay as a New Year’s gift. As Kepler wrote on the title page:
To the honorable Counselor at the Court of his Imperial Majesty, Lord Matthaus Wacker von Wackenfels, a Decorated Knight and Patron of Writers and Philosophers, my Lord and Benefactor.
As the title suggests, Kepler’s main concern was the question of why snowflakes are almost always six-pointed…
Follow the train of thought from the stacking of spheres to the intricacies of tiling at “Snowflakes and Cannonball Stacks.”
* Orhan Pamuk, Snow
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As we pause to ponder patterns, we might recall that it was on this date in 1891, about 20 miles outside of Midland, Texas, that the first rainmaking experiment in the U.S. was conducted. Robert St. George Dyrenforth, a Washington patent attorney and retired Army officer, led a team that used “mortars, casks, barometers, electrical conductors, seven tons of cast-iron borings, six kegs of blasting powder, eight tons of sulfuric acid, one ton of potash, 500 pounds of manganese oxide, an apparatus for making oxygen and another for hydrogen, 10- and 20-foot-tall muslin balloons and supplies for building enormous kites” to create enormous explosions meant to help clouds form. Their efforts– which were based more on Dyrenforth’s instinct than on anything resembling scientific evidence– were entirely unsuccessful. Still, at a time of extreme drought, it’s likely that almost anything seemed worth trying. (The full– and very entertaining– story, here.)
“Geographers never get lost. They just do accidental field work.”*…
A Facebook friend recently noted that Turkey was “a remarkably rectangular country.” I wondered how it compared to other countries, and this post shows my answers (Turkey is 15th; Egypt is the most rectangular; full table below). I defined the rectangularness of a country as its maximum percentage overlap with a rectangle of the same area, working in the equirectangular projection (i.e., x = longitude, y = latitude). Ideally each country would get its own projection, but equirectangular rectangles feel at least linguistically thematic and are easier to code…
David Barry‘s ranking of “The rectangularness of countries.”
* Nicholas Chrisman, Professor of Geomatic Sciences, Université Laval
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As we get square, we might send paradigm-shaping birthday greetings to a woman who enabled mapping of an altogether different– and world-changing– sort: Rosalind Franklin; she was born on this date in 1920. A biophysicist and X-ray crystallographer, Franklin captured the X-ray diffraction images of DNA that were, in the words of Francis Crick, “the data we actually used” when he and James Watson developed their “double helix” hypothesis for the structure of DNA. Indeed, it was Franklin who argued to Crick and Watson that the backbones of the molecule had to be on the outside (something that neither they nor their competitor in the race to understand DNA, Linus Pauling, had understood). Franklin never received the recognition she deserved for her independent work– her paper was published in Nature after Crick and Watson’s, which barely mentioned her– and she died of cancer four years before Crick, Watson, and their lab director Maurice Wilkins won the Nobel Prize for the discovery.
“Math is sometimes called the science of patterns”*…
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From Katie Steckles, help for the Holidays…
Special Holiday bonus: the story behind those massive bows that bedeck cars given as Holiday presents.
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As we fold with care, we might recall that it was on this date in 1937 that Walt Disney released the first full-length animated feature film produced in the U.S. (and the first produced anywhere in full color), Snow White and the Seven Dwarfs.
The original theatrical one-sheet
“You can’t criticize geometry. It’s never wrong.”*…
In the world of mathematical tiling, news doesn’t come bigger than this. In the world of bathroom tiling – I bet they’re interested too.
If you can cover a flat surface using only identical copies of the same shape leaving neither gaps nor overlaps, then that shape is said to “tile the plane.” Every triangle can tile the plane. Every four-sided shape can also tile the plane.
Things get interesting with pentagons. The regular pentagon cannot tile the plane. (A regular pentagon has equal side lengths and equal angles between sides, like, say, a cross section of okra, or, erm, the Pentagon). But some non-regular pentagons can.
The hunt to find and classify the pentagons that can tile the plane has been a century-long mathematical quest, begun by the German mathematician Karl Reinhardt, who in 1918 discovered five types of pentagon that do tile the plane…
Pentagons remain the area of most mathematical interest when it comes to tilings since it is the only of the ‘-gons’ that is not yet totally understood…
Read the whole story– and see all 15 types of pentagonal tilings discovered so far– at “Attack on the pentagon results in discovery of new mathematical tile.”
* Paul Rand
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As we grab the grout, we might recall that it was on this date in 1953, after a year of experimentation, that marine engineer and retired semi-pro baseball player David Mullany, Sr. invented the Wiffleball. (He patented it early the following year.) Watching his 13-year-old son play with a broomstick and a plastic golf ball ball in the confines of their backyard, Mullany worried that the effort to throw a curve would damage his young arm. So he fabricated a full- (baseball-)sized ball from the plastic used in perfume packaging, with oblong holes on one side… a ball that would naturally curve. The balls had the added advantage, given their light weight, that they’d not break windows.
David Jr. came up with the name: he was fond of saying that he had “whiffed” the batters that he struck out with his curves. The “h” was dropped, the name trademarked, and (after Woolworth’s adopted the item) a generation of young ballplayers– and their parents– converted.
“Geometry is not true, it is advantageous”*…
The tesseract is a four dimensional cube. It has 16 edge points v=(a,b,c,d), with a,b,c,d either equal to +1 or -1. Two points are connected, if their distance is 2. Given a projection P(x,y,z,w)=(x,y,z) from four dimensional space to three dimensional space, we can visualize the cube as an object in familar space. The effect of a linear transformation like a rotation
| 1 0 0 0 | R(t) = | 0 1 0 0 | | 0 0 cos(t) sin(t) | | 0 0 -sin(t) cos(t) |in 4d space can be visualized in 3D by viewing the points v(t) = P R(t) v in R3.
* Henri Poincare
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As we follow the bouncing ball, we might spare a thought for Felix Klein; he died on this date in 1925. A mathematician of broad gauge, he is best remembered for his work in non-Euclidean geometry, perhaps especially for his work on synthesis of geometry as the study of the properties of a space that are invariant under a given group of transformations, known as the Erlanger Program, which profoundly influenced mathematics. He created the Klein bottle, a one-sided closed surface–a non-orientable surface with no inside and no outside– that cannot be constructed in Euclidean space.
A Klein bottle
