I turn 49 today. This is a 49-pointed star, to mark the occasion — an old tradition of mine.
My first birthday was in 1968 — on the day I was born. A year later, on my second birthday, I turned one year old. Carry this up to the present, and that makes today, when I turn 49, my 50th birthday. For those reasons, here’s a second star, with 50 points.
There are objects in hyperspace known as duoprisms, which have prismatic cells. This one’s cells are 24 dodecagonal prisms. It was made using Stella 4d, available here.
The faces of this polyhedron are 12 regular pentagons, 20 equilateral triangles, 30 rhombi, and 60 almost-square trapezoids. I created it with Stella 4d, which is available at this website.
Stella 4d was used to make this image, and you may try it for free by following this link.
Of the thirty faces of this polyhedron, only the yellow parallelograms are not pentagons. I used Stella 4d to make this; you may try it for free at http://www.software3d.com/Stella.php.
The polyhedron above is the 12th stellation of the truncated cube. The one below is the 14th.
The next one shown is the 18th and final stellation. If stellated again, the result is an ordinary truncated cube.
These virtual models were made using Stella 4d, software you may try for yourself at http://www.software3d.com/Stella.php.
Truncated octahedra are among the special polyhedra which can fill space without leaving any gaps. There are others, as well. This image was created using Stella 4d, software you may try, for yourself, right here. There is a free “try it before you buy it” download available.
This tessellation appears, at first, to be regular (or “Archimedean,” by analogy with the Archimedean solids), for all of the polygons included are regular. However, it is not vertex-transitive, which keeps it from qualifying as a regular or Archimedean tessellation.
This is the truncated cube, one of the thirteen Archimedean solids.
If the truncation-planes are shifted, and increased in number, in just the right way, this variation is produced. Its purple faces are regular dodecagons, and the orange faces are kites — two dozen, in eight sets of three.
Applying yet another truncation, of a specific type, produces the next polyhedron. Here, the regular dodecagons are blue, and the red triangles are equilateral. The yellow triangles are isosceles, with a vertex angle of ~41.4 degrees.
All three of these images were produced using Stella 4d, available at this website.
The yellow faces of this polyhedron are parallelograms, while the red ones are trapezoids. To demonstrate its chirality, here is the compound of it, and its own mirror-image.
Both of these “virtual polyhedra” were made using Stella 4d: Polyhedron Navigator, a program available at this website. It has a free trial download available.
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