Camping In Hexagonal Tents On a Tessellated Plane

Image

Camping In Hexagonal Tents On a Tessellated Plane

Squares and Triangles

Image

Squares and Triangles

Here’s what it looks like with different colors (click to enlarge it):

tess with sq & tri 2

A Gallery of Nine Tessellations Using Hexagons

Image

hextess

Pictured above is the most familiar hexagonal tessellation. I’ve found some additional tessellations which use equilateral (but non-equiangular) hexagons, and have radial symmetry. They appear, using various coloring-schemes, below.

Hex radial tessellationHex radial tessellation 2Hex radial tessellation 3Hex radial tessellation 4radial octagonal mandala 2radial octagonal mandala 2Buntitleduntitled ic

A Rhombic Dodecahedron Featuring Rotating Pentagonal Mandalas

Image

A Rhombic Dodecahedron Featuring Rotating Pentagonal Mandalas

Geometer’s Sketchpad and MS-Paint were both used to make the images on the faces of this polyhedron, and then Stella 4d was used to put it all together and create this rotating image. Stella may be bought, and/or tried for free, at www.software3d.com/Stella.php.

A Pentagonal Mandala In Primary Colors

Image

A Pentagonal Mandala In Primary Colors

The Center of a Radial Tessellation Featuring Regular Pentadecagons

Image

The Center of a Radial Tessellation Featuring Regular Pentadecagons

A Gallery of 27 Polyhedra with Cuboctahedral Symmetry

Image

Some of these polyhedra have “normal” cuboctahedral symmetry, while others have the chiral variant of that symmetry-type — in other words, the same type of symmetry found in the snub cube.

Some Polyhedra with Cuboctahedral Symmetry

CO1Convex hullCO2 Convex hullCconvex hullDual of Convex hullhexagons and squaresStellated Convex hullStellated Convex hulllAugmented Stellated Convex hullCoonvex hullConvex hull of prism-augmented snub cubedual of Coonvex hulllreaugmented dual of Convex hull of prism-augmented snub cubeStellated Polydual of Convex hull of prism-augmented snub cubeanother augmentation of dual of Convex hull of prism-augmented snub cubedual of Coonvex hullFaceted Convex hullStellated Poly2Stellated Poly3Faceted Convex hulllStellated Poly4Dual of Convex hulglDual of Convex hugFaceted Convex hulfgdsgthxlFaceted Convex hulfgdsgthgfsxlFaceted Convex hufdlfgdsgthgfsxl

I used Stella 4d:  Polyhedron Navigator to make these images, and you can find that program at http://www.software3d.com/Stella.php.

Thirty Flying Rhombi

Image

Sixty Flying Rhombi

I used Stella 4d to make this image, and you can find that program at http://www.software3d.com/Stella.php.

Thirty Flying Hexagons

Image

Sixty Flying Hexagons

When I tweet a link to this blog-post as @robertlovespi, it will likely be quickly retweeted by @hexagonbot, simply because of the term “hexagon” being included in the tweet. What I don’t understand: Why do other polygons not have bots of their own?

I used Stella 4d to make this image, and you can find that program at http://www.software3d.com/Stella.php.

Some Recommeded Websites for Polyhedral Enthusiasts

There are a lot of websites devoted to polyhedra. Here are some of the best.

This one is run by Jonathan Bowers:  http://www.polytope.net/hedrondude/home.htm.

Here’s the portal-page to George Hart’s pages, with links to a LOT of cool stuff he’s made: http://www.georgehart.com/.

I don’t know who runs this one:  http://polyhedra.org/poly/

This one is Robert Webb’s. He’s the person who wrote Stella 4d, the program I use most often on my own blog to create polyhedral images: http://www.software3d.com/Gallery.php. (Also, while we do share a first name, he is not the same person as me, as a few readers of my blog have thought in the past.)

Craig Kaplan has a page of links to other pages of his, right here: http://www.cgl.uwaterloo.ca/~csk/projects/. Of those, my favorites are the sections on John solid near-misses (http://www.cgl.uwaterloo.ca/~csk/projects/nearmisses/), as well as on symmetrohedra:  http://www.cgl.uwaterloo.ca/~csk/projects/symmetrohedra/

Jim McNeill’s polyhedra site is here:  http://www.orchidpalms.com/polyhedra/

Here’s a good one, but I don’t know its creator’s full name: http://colinspics.org/index.htm

Finally, one by Vladimir Bulatov:  http://www.bulatov.org/polyhedra/index.html

This is definitely not a complete list. If you know of other good polyhedron-oriented websites, please leave links to them in a comment on this post.