Tessellation of Concave Hexagons
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Tag Archives: hexagon
A Variant of the Icosidodecahedron Featuring Four Hexagons
This was made using Stella 4d, a program you can try for yourself at http://www.software3d.com/Stella.php.
A Variant of the Rhombicosidodecahedron Featuring Four Hexagons
This was made using Stella 4d, a program you can try for yourself at http://www.software3d.com/Stella.php.
Tessellation of Blue Triangles and Yellow Concave Pentagons
Alternately, this can be seen as a tessellation of blue diconcave hexagons and yellow triconcave enneagons. Which do you see?
Tessellation Featuring Regular Enneagons, Regular Hexagons, and Isosceles Triangles
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A Mandala Made of Hexagons, Enneagons, and Dodecagons
I recently re-discovered this “lost work,” which I made using Geometer’s Sketchpad, in 2011 — before I started this blog, which is why it has not appeared here before.
Order-Six Radial Tessellations of the Plane, Using Elongated and Equilateral Hexagons, Rendered with Twelve Different Coloring-Schemes
I explored radial tessellations of the plane, using only hexagons, in this earlier post. Order-three tessellations of this type are the familiar regular-hexagon tessellations of the plane. With higher-order all-hexagon radial tessellations, though, the hexagons must be elongated, although they can still remain equilateral, and all congruent, with bilateral symmetry. In that previous post, examples were shown of order 4, 5, and 8, in addition to the familiar order-3 regular-hexagon tessellation.
This left out order-6, of which I show many examples below. As it turns out, this particular radial tessellation lends itself particularly well to a variety of coloring-schemes. In the first picture, the construction-circles, -points, and -lines I used are shown; in the rest, they are hidden.
No upper limit exists to the order-number of these all-hexagon radial tessellations — although the larger that number gets, the thinner the hexagons become, relative to their edge length. At some point (which I expect would vary from person to person), as the order-number increases, the hexagons needed will become so thin that they will no longer be recognizable as hexagons.
Next, with construction artifacts hidden, are some two-color designs I found.
Here are some which use three colors each:
I also found some four-color patterns with interesting symmetry:
Finally, here are some which each use six colors.
A Polyhedron Featuring Octagons, Hexagons, and Pentagons, All Irregular
I made this using Stella 4d, software which you can try for free right here.
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A Polyhedron with Eight Regular Hexagonal Faces and Twenty-Four Irregular Pentagonal Faces
There are many polyhedra that include only hexagons and pentagons as faces — infinitely many, in fact. Most of the well-studied ones include twelve regular pentagonal faces, though, but this polyhedron has twenty-four pentagons, none of which are regular, in six panels of four pentagons each. Its regular faces are the eight hexagons, in the face-planes of the faces of an octahedron.
I made this with Stella 4d, software you can try here.
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