Pedal to the Medal

“Once, in a contest with a rival, he painted a blue curve on a huge sheet of paper. Then he dipped the feet of a chicken in red paint and persuaded the bird to walk all over the paper. The resulting image, he said, represented the Tatsuta river with red maple leaves floating in it. The judge gave him the prize.” — The Japanese artist Katsushika Hokusai (c. 1760-1849) described in Thomas W. Hodgkinson’s and Hubert van den Bergh’s How to Sound Cultured (2015).

Oneiric Ocean

20000-leagues-under-the-sea


I like this illustration of a scene in Jules Vernes’s Twenty Thousand Leagues Under the Sea (1870) even more because it has at least one mistake in it. At least, I think it’s a mistake: the jellyfish on the upper left are two Portuguese men-o’-war (really colonial hydrozoans, not jellyfish). They have gas-filled float-bladders, so in reality you see them only on the surface, not hanging in midwater like that. The mistake makes the scene like a dream. The absence of colour is good too: it fixes the illustration firmly in the past and the colours you imagine are more vivid. The artist is imagining, dreaming, conjuring a vision of an oneiric ocean.

Rock’n’Roll Suislide

Q. Each face of a convex polyhedron can serve as a base when the solid is placed on a horizontal plane. The center of gravity of a regular polyhedron is at the center, therefore it is stable on any face. Irregular polyhedrons are easily constructed that are unstable on certain faces; that is, when placed on a table with an unstable face as the base, they topple over. Is it possible to make a model of an irregular convex polyhedron that is unstable on every face?

Portrait of Luca Pacioli (1495)

Portrait of Luca Pacioli (1495)


A. No. If a convex polyhedron were unstable on every face, a perpetual motion machine could be built. Each time the solid toppled over onto a new base it would be unstable and would topple over again.

 — From “Ridiculous Questions” in Martin Gardner’s Mathematical Magical Show (1965), chapter 10.

Performativizing Papyrocentricity #46

Papyrocentric Performativity Presents:

Machina MundiThe Invention of Science: A New History of the Scientific Revolution, David Wootton (Allen Lane 2015)

Wandering WondersPlankton: Wonders of the Drifting World, Christian Sardet (The University of Chicago Press 2015)

Love BuzzA Buzz in the Meadow, Dave Goulson (Jonathan Cape 2014)

Quake’s ProgressThe Million Death Quake: The Science of Predicting Earth’s Deadliest Natural Disaster, Roger Musson (Palgrave Macmillan 2012)

Sin after CinGargoyle Girls from Beelzebub’s Ballsack: The Sickest, Sleaziest, Splanchnophagousest Slimefests in Scum Cinema, Dr Joan Jay Jefferson (TransToxic Texts 2016)


Or Read a Review at Random: RaRaR

The Art Grows Onda

Anyone interested in recreational mathematics should seek out three compendiums by Ian Stewart: Professor Stewart’s Cabinet of Mathematical Curiosities (2008), Professor Stewart’s Hoard of Mathematical Treasures (2009) and Professor Stewart’s Casebook of Mathematical Mysteries (2014). They’re full of ideas and puzzles and are excellent introductions to the scope and subtlety of maths. I first came across Alexander’s Horned Sphere in one of them. I also came across this simpler shape that packs infinity into a finite area:

unicorn_triangle

I call it a horned triangle or unicorn triangle and it reminds me of a wave curling over, like Katsushika Hokusai’s The Great Wave off Kanagawa (c. 1830) (“wave” is unda in Latin and onda in Spanish).

The Great Wave off Kanagawa by Katsushika Hokusai (1760–1849)

The Great Wave off Kanagawa by Katsushika Hokusai (1760–1849)

To construct the unicorn triangle, you take an equilateral triangle with sides of length 1 and erect a triangle with sides of length 0.5 on one of its corners. Then on the corresponding corner of the new triangle you erect a triangle with sides of length 0.25. And so on, for ever.

unicorn_multicolor

unicorn_animated

When you double the sides of a polygon, you quadruple the area: a 1×1 square has an area of 1, a 2×2 square has an area of 4. Accordingly, when you halve the sides of a polygon, you quarter the area: a 1×1 square has an area of 1, a 0.5 x 0.5 square has an area of 0.25 or 1/4. So if the original triangle of the unicorn triangle above has an area of 1 rather than sides of 1, the first triangle added has an area of 0.25 = 1/4, the next an area of 0.0625 = 1/16, and so on. The infinite sum is this:

1/4 + 1/16 + 1/256 + 1/1024 + 1/4096 + 1/16384…

Which equals 1/3. This becomes important when you see the use made of the shape in Stewart’s book. The unicorn triangle is a rep-tile, or a shape that can be divided into smaller copies of the same shape:

unicorn_reptile_static

unicorn_reptile

An equilateral triangle can be divided into four copies of itself, each 1/4 of the original area. If an equilateral triangle with an area of 4 is divided into three unicorn triangles, each unicorn has an area of 1 + 1/3 and 3 * (1 + 1/3) = 4.

Because it’s a rep-tile, a unicorn triangle is also a fractal, a shape that is self-similar at smaller and smaller scales. When one of the sub-unicorns is dropped, the fractals become more obvious:

unicorn_fractal1


unicorn_fractal2


unicorn_fractal3


Elsewhere other-posted:

Rep-Tiles Revisited