#knottys math

Einstein monotiles as bead rags.

The spherical beads are packed densely to form a sensorically interesting surface, materialized as a "rag".

Assembled in the shape of the Einstein monotile, they can be played with as if they are mosaic bricks.

And as you can see, these three monotile rags tile relatively neatly. There are only gaps between the tiles where I would have needed half a bead.

This has to do with the angles of the monotile:

The einstein monotiles have angles of 60, 120, 240, 90 and 270 degrees.

"Flat hat on a flat cat":

For that Kepler-Poinsot-Polyhedron I started to draw an icosahedron and continued to add triangular pyramids as "hats" on top of each of the 20 triangular icosahedron faces.

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This is the Small Stellated Dodecahedron I drew a while back:

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How are the Dodecahedron, Icosahedron, Small and Great Stellated Dodecahedron related?

(Source)

Truncation of the Small Stellated Dodecahedron:

Truncation of the Great Stellated Dodecahedron:

Front: 5 platonic solids and the 13 Archimedean solids

Back: Root-like structures, depicting some relationships between these 18 solids.

Both simultaneously (Holding it against light source):

By cutting off the vertices of the dodecahedron you get the truncated dodecahedron.

This results in the 12 pentagonal faces of the dodecahedron turning into 12 dekagonal (10-gonal) and 20 triangular faces.

Today I finished this friend-shaped archimedian solid.

I drew the tiny drawing in the down right corner of the card and resized it with my printer.

The drawing:

I drew this Truncated cuboctahedron in an isometric projection and used my beloved isometric dot paper.

To start with the truncated cuboctahedron I started to draw a cube (with pencil).

Then I altered the cube by drawing a cuboctahedron in it (with pencil as well). I truncated th vertices of the cube like in the depiction below:

Then I altered the cuboctahedron drawing with another truncation - resulting in the truncated cuboctahedron shape.

[For clarification: I later erased the pencil lines of the cube and cuboctahedron, because it became messy and these lines were just there to help in the drawing process.

(^front view, v back view)

Front and back view simultaneously:

The pentagonal icositetrahedron is the dual polyhedron of the snub cube.