horned triangle | Overlord In Terms of Core Issues Around Maximal Engagement with Key Notions of the Über-Feral

Pre-previously on Overlord-in-terms-of-Core-Issues-around-Maximal-Engagement-with-Key-Notions-of-the-Über-Feral, I interrogated issues around this shape, the horned triangle:

Horned Triangle (more details)

Now I want to look at the tricorn (from Latin tri-, “three”, + -corn, “horn”). It’s like a horned triangle, but has three horns instead of one:

Tricorn, or three-horned triangle

These are the stages that make up the tricorn:

Tricorn (stages)

Tricorn (animated)

And there’s no need to stop at triangles. Here is a four-horned square, or quadricorn:

Quadricorn

Quadricorn (animated)

Quadricorn (coloured)

And a five-horned pentagon, or quinticorn:

Quinticorn, or five-horned pentagon

Quinticorn (anim)

Quinticorn (col)

And below are some variants on the shapes above. First, the reversed tricorn:

Reversed Tricorn

Reversed Tricorn (anim)

Reversed Tricorn (col)

The nested tricorn:

Nested Tricorn (anim)

Nested Tricorn (col)

Nested Tricorn (red-green)

Nested Tricorn (variant col)

The nested quadricorn:

Nested Quadricorn (anim)

Nested Quadricorn

Nested Quadricorn (col #1)

Nested Quadricorn (col #2)

Finally (and ferally), the pentagonal octopus or pentapus:

Pentapus (anim)

Pentapus

Pentapus #2

Pentapus #3

Pentapus #4

Pentapus #5

Pentapus #6

Pentapus (col anim)

Elsewhere other-engageable:

• The Art Grows Onda — the horned triangle and Katsushika Hokusai’s painting The Great Wave off Kanagawa (c. 1830)

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:

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)

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.

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:

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

Overlord In Terms of Core Issues Around Maximal Engagement with Key Notions of the Über-Feral

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Tag Archives: horned triangle

Horn Again

The Art Grows Onda