#regular polyhedron

[art-of-mathematics]

In this perspective the icosahedron drawing fits neatly into this hexagon shape.

[art-of-mathematics]

Hyper-tetrahedron Tetrahedral prism model

That is a model I made back in ~2013 with paper clips.

Back then I attempted to build tiny chain reactions (marble tracks) at the wall of my room. I remember building a cube-like marble releaser using only sticks and tie. - Cubes are not stable like tetrahedra when only using sticks for the main edges. Without additional edges the cube could be distorted by torsion. This made it annoying as it resulted in needing far more sticks to stabilize. -> Tetrahedra are more efficient in these aspects.

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The molecular structure of diamonds is also tetrahedral. (It could also explain why diamonds are so hard/stable. )

- - - (correction after posting) - - -

hyper-tetrahedron/5-cell/4-simplex (Left) and tetrahedral prism (Right):

[art-of-mathematics]

Icosahedron net - "coiling up"

This icosahedron net is really cool, as it is just like a "string of triangles" you can just roll up to form the icosahedron.

Result:

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The hexagon grid is really helpful for drawing equilateral triangles.

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It reminds me of this:

Diffraction - Source: German PDF [found at TU-dresden website]

...

[art-of-mathematics]

Self-similar tetrahedra packing with 4 smaller tetrahedra and 1 octahedron

That was my contemplation some months ago:

Today I started a trashy/bad-quality prototype 3D model with paper - and it appears like a first step for a proof to my contemplation:

- 4 tetrahedra with side length of 3 cm

- 1 tetrahedron with side length of 6 cm

- 1 octahedron with side length of 3cm

-> the 4 small tetrahedra and the octahedron packed together fill the large tetrahedron - wonderfully packed 3D space. There are no gaps. (A better quality model might depict it better than this trashy model. )

Next step: I will craft a better quality 3D model.

[art-of-mathematics]

Icosahedron

Today I improved the thread connections.

[art-of-mathematics]

Icosahedron net - slightly colorful visualization

Currently I am obsessed with polyhedra nets - and crafting tiny models and creating visualizations.

For this one I created a template for the net that somehow reminds me of the cornu spiral - as this kind of net can be "coiled up" to fold the polyhedron.

Furthermore, I used 12 colors for the 12 vertices - in a gradient-like arrangement (it starts at red and gradually follows the route at orange, yellow, green, blue, violet, and ends at pink.)

[art-of-mathematics]

Icosahedron model

The sketch/doodle/idea on paper:

My model has a side length of a = 5cm.

The result:

Ingredients:

- The icosahedron net (uncoiled): [Analogy terms: Red part is the "top", blue part the "bottom" and the yellow part is the "belt".]

- three rectangles with ratio 1 to Phi (~1.618) put together: [Hence, a = 5cm and b = 5cm*1.618 = ~ 8cm]

Assembly

[art-of-mathematics]

Bought that 'stim ball', because my affinity for regular polyhedra was too strong to resist the urge to buy.

Actually it is declared as an object you should put in a washing machine for better drying effects. But I instantly saw the many other things I could use it for, like as a stim toy. And a dodecahedron model. Lol