Two Non-convex Polyhedra with Cuboctahedral Symmetry

Posted on 16 January 2019 by RobertLovesPi

I made both of these while playing around with Stella 4d: Polyhedron Navigator, a program you can try out for free at http://www.software3d.com/Stella.php.

A Gallery of Fourteen Polyhedra with Cuboctahedral Symmetry

Posted on 4 November 2016 by RobertLovesPi

Any of these rotating polyhedra may be made larger with a single click. All were created using Stella 4d, a program you may try for free at this website: http://www.software3d.com/Stella.php.

A Tetrahedral Exploration of the Icosahedron

Posted on 24 June 2016 by RobertLovesPi

Mathematicians have discovered more than one set of rules for polyhedral stellation. The software I use for rapidly manipulating polyhedra (Stella 4d, available here, including as a free trial download) lets the user choose between different sets of stellation criteria, but I generally favor what are called the “fully supported” stellation rules.

For this exercise, I still used the fully supported stellation rules, but set Stella to view these polyhedra as having only tetrahedral symmetry, rather than icosidodecahedral (or “icosahedral”) symmetry. For the icosahedron, this tetrahedral symmetry can be seen in this coloring-pattern.

The next image shows what the icosahedron looks like after a single stellation, when performed through the “lens” of tetrahedral symmetry. This stellation extends the red triangles as kites, and hides the yellow triangles from view in the process.

The second such stellation produces this polyhedron — a pyritohedral dodecahedron — by further-extending the red faces, and obscuring the blue triangles in the process.

The third tetrahedral stellation of the icosahedron produces another pyritohedral figure, which further demonstrates that pyritohedral symmetry is related to both icosidodecahedral and tetrahedral symmetry.

The fourth such stellation produces a Platonic octahedron, but one where the coloring-scheme makes it plain that Stella is still viewing this figure as having tetrahedral symmetry. Given that the octahedron itself has cuboctahedral (or “octahedral”) symmetry, this is an increase in the number of polyhedral symmetry-types which have appeared, so far, in this brief survey.

Next, I looked at the fifth tetrahedral stellation of the icosahedron, and was surprised at what I found.

While I was curious about what would happen if I continued stellating this polyhedron, I also wanted to see this fifth stellation’s convex hull, since I could already tell it would have only hexagons and triangles as faces. Here is that convex hull:

For the last step in this survey, I performed one more tetrahedral stellation, this time on the convex hull I had just produced.

A Gallery of Cuboctahedral Polyhedra, and Polyhedral Compounds, Some of Them Chiral

Posted on 11 March 2016 by RobertLovesPi

Any of these images may be enlarged with a click.

Compound of Trunc Octa and Trunc Cube in 4colors

Zonohedrified Trunc Octa not zonish based on faces and vertices

314 faces includes hexadecagon and enneagons

compound of four polyhedra featuring kites

Compound of trun octa and trun cube by component

24 pentagons and 6 octagons chiral cuboctahedral and convex

56 triangles chiral and cuboctahedral and convex

Compound of triakis octa and tetrakis cube

86 faces 6 octagons 8 green hexagons 24 each of two other types of hexagons 24 pentagons

They were all created using Stella 4d, available at this website.

Eleven Convex, Non-Chiral Polyhedra Featuring Cuboctahedral Symmetry

Posted on 23 January 2016 by RobertLovesPi

To enlarge any of these images, simply click on the ones you choose.

Symmetrohedra formed by prism-augmentation of the enneagonal near-miss THEN created convex hull THEN used TTMFR

regular tetrakaiicosagons and triangles and three types of trapezoid

6 reg 24-gons 8 irreg hexagons 24 each of two types of trapezoid 12 very narrow rectangles 72 faces in all

All of these images were created using Stella 4d: Polyhedron Navigator, available at http://www.software3d.com/Stella.php.

87 Rotating Non-Convex, Non-Chiral Polyhedral Images Featuring Icosidodecahedral Symmetry, Plus Four Which Snuck In with Cuboctahedral Symmetry — Can You Find All the Intruders?

Posted on 22 January 2016 by RobertLovesPi

To see larger versions of any of these, simply click on the images.

Excavated GRID with triangular cupola and more embellishments

Dodecahedra 5 compound with icosahedra 5

compound great icosahedron and dodecahedron

Compound of Trunc Icosidodeca and Rhombic Triaconta

207th Stellateion of the Rhombicosidodeca converted to base model and colored by face type

Faceted Dodeca and should be a uniform polyhedron

Faceted Icosidodeca six reg decagons going through center twelve star pentagons 30 golden rectangles nth stellation more n and more and more

Faceted Icosidodeca six reg decagons going through center twelve star pentagons 30 golden rectangles nth stellation more n and more

Faceted Icosidodeca six reg decagons going through center twelve star pentagons 30 golden rectangles nth stellation more n

Faceted Icosidodeca golden rectangles dodecagons and star pentagons

24 to this point….

Faceted Stefdhllated Rhgdhombic Triaconta

Faceted Stellahhctejhfgfd Facjhfjhfeted Dual

Faceted Icosidodeca six reg decagons going through center twelve star pentagons 30 golden rectangles nth stellation

Faceted Stefdhllated Rhgdhombic Triaconta base dual compound

Faceted Triakisicosa which is the dual of the 22nd stellation of the truncated dodecahedron

Faceted truncated icosahedron the dual made of sixty intersecting and irregalar hexagons

Facetedfdjsjh Shgdytelhgytlated Faceted Dual

That’s 40 so far…

$fractured great dodecahedron$

great rhombicosidodecahedral shell produced by drilling

Now the count is at four dozen.

That was 26 more, so there are 48 + 26 = 74 so far.

stellated icosa final stell dual a faceting of the RTC rcm

stellated icosa final stell dual a faceting of the RTC

Stellated Icosidodeca fs colored by i dunno

Faceted Rhombic Triaconta with Four Faces Meeting At an Edge

Faceted Trunc Icosidodeca COMPOUND 15 CUBOIDS FIVE-COLOR VERSION DUAL

Faceted Truncated Icosidodecahedron / Compound of Fifteen Cuboids

Now the count is up to 83.

Faceted Poly GRID compound of six decagonal prisms

Faceted Rhombicosidodeca compound of six elongated pentagonal prisms the dual six trapezohedra

Faceted Rhombicosidodeca compound of six elongated pentagonal prisms

Faceted Trunc Icosidodeca COMPOUND 15 CUBOIDSCOLORED BY FACE TYPE DUAL

Compound of enantiomorphic pair of faceted snub dodecah

34th stellation of the Small Ditri Icosidodeca

So there were 91 of these stored on my hard drive, from all my “hard work” playing with polyhedra using Stella 4d: Polyhedron Navigator. (It will be good for my computer to get all that hard drive space back!) If you’d like to try playing with the same program — for free — just try the free download at http://www.software3d.com/Stella.php.

Six Non-Convex Polyhedra with Cuboctahedral Symmetry

Posted on 18 January 2016 by RobertLovesPi

Each of these polyhedral images (any of which may be enlarged with a click) was created using Stella 4d: Polyhedron Navigator, and this program may be tried for free at http://www.software3d.com/Stella.php.

Also, a question, for regular readers of my blog — you have probably noticed that this post has a different format, but it’s just an experimental thing I’m trying out.

Do you prefer this style of polyhedra-post, or the format I usually use?

Two Polyhedral Compounds: the Icosidodecahedron with the Truncated Cube, and the Rhombic Triacontahedron with the Triakis Octahedron

Posted on 22 May 2015 by RobertLovesPi

These two compounds, above and below, are duals. Also, in each of them, one polyhedron with icosidodecahedral symmetry is combined with a second polyhedron with cuboctahedral symmetry to form a compound with pyritohedral symmetry: the symmetry of a standard volleyball.