From the Rhombic Enneacontahedron to an All-Kite Polyhedron

Posted on 15 February 2020 by RobertLovesPi

This is the rhombic enneacontahedron, one of the few well-known zonohedra. Its ninety faces have two types: sixty wide rhombi, and thirty narrow rhombi.

In the image above, the thirty narrow rhombi of the rhombic enneacontahedron have been augmented with prisms.

The next step in today’s polyhedral play was to create the convex hull of this augmented rhombic enneacontahedron. This produced the solid shown immediately above. To make the one shown below, I next used a function called “try to make faces regular.” The result is a symmetrohedron with 122 faces: 12 regular pentagons, 30 rhombi, 60 almost-square isosceles trapezoids, and thirty equilateral triangles.

Finally, I examined the dual of this symmetrohedron, which turned out to have 120 faces: two sets of sixty kites each.

The program I used to create these polyhedral images is called Stella 4d, and you can try it yourself (as a free trial download) at http://www.software3d.com/Stella.php.

A Polyhedron Made of Regular Dodecagons and Kites

Posted on 3 November 2019 by RobertLovesPi

Made with Stella 4d, available as a free trial download at http://www.software3d.com/Stella.php.

A Polyhedron Featuring Eight Regular Enneagons and Twenty-four Kites

Posted on 19 November 2018 by RobertLovesPi

8 enneagons and 24 kites

I made this using Stella 4d, which you can try for free right here.

A Chiral Polyhedron Made of Kites and Triangles, Along with Its Dual, Made of Triangles and Isosceles Trapezoids

Posted on 2 July 2018 by RobertLovesPi

To make this polyhedron using Stella 4d (available here), one starts with the icosahedron, drops the symmetry of the model down from icosahedral to tetrahedral, and then stellates it once. The result is a chiral solid featuring four triangular faces and twelve kites:

The dual of this polyhedron, which is also chiral, has four triangular faces, and twelve faces which are isosceles trapezoids. It is a type of faceted dodecahedron — a partial faceting, meaning it is made without using all of the dodecahedron’s vertices.

Eight Kite-Rhombus Solids, Plus Five All-Kite Polyhedra — the Convex Hulls of the Thirteen Archimedean-Catalan Compounds

Posted on 29 April 2018 by RobertLovesPi

In a kite-rhombus solid, or KRS, all faces are either kites or rhombi, and there are at least some of both of these quadrilateral-types as faces. I have found eight such polyhedra, all of which are formed by creating the convex hull of different Archimedean-Catalan base-dual compounds. Not all Archimedean-Catalan compounds produce kite-rhombus solids, but one of the eight that does is derived from the truncated dodecahedron, as explained below.

The next step is to create the compound of this solid and its dual, the triakis icosahedron. In the image below, this dual is the blue polyhedron.

The convex hull of this compound, below, I’m simply calling “the KRS derived from the truncated dodecahedron,” until and unless someone invents a better name for it.

The next KRS shown is derived, in the same manner, from the truncated tetrahedron.

Here is the KRS derived from the truncated cube.

The truncated icosahedron is the “seed” from which the next KRS shown is derived. This KRS is a “stretched” form of a zonohedron called the rhombic enneacontahedron.

Another of these kite-rhombus solids, shown below, is based on the truncated octahedron.

The next KRS shown is based on the rhombcuboctahedron.

Two of the Archimedeans are chiral, and they both produce chiral kite-rhombus solids. This one is derived from the snub cube.

Finally, to complete this set of eight, here is the KRS based on the snub dodecahedron.

You may be wondering what happens when this same process is applied to the other five Archimedean solids. The answer is that all-kite polyhedra are produced; they have no rhombic faces. Two are “stretched” forms of Catalan solids, and are derived from the cuboctahedron and the icosidodecahedron:

all kites based on the cuboctahedron also a stretching of the strombic icositetrahedron which is the dual of the rhombcuboctahedron

all kites based on the icosidodechadedron and a stretching of the strombic hexecontahedron which is the dual of the RID

If this procedure is applied to the rhombicosidodecahedron, the result is an all-kite polyhedron with two different face-types, as seen below.

The two remaining Archimedean solids are the great rhombcuboctahedron and the great rhombicosidodecahedron, each of which produces a polyhedron with three different types of kites as faces.

all kites based on the great rhombcuboctahedron

The polyhedron-manipulation and image-production for this post was performed using Stella 4d: Polyhedron Navigator, which may be purchased or tried for free at http://www.software3d.com/Stella.php.

Shield Mandala

Image

shield-mandala

Two Dozen Kites Each, in Three Sets, as Faces of a Polyhedron

Posted on 22 October 2016 by RobertLovesPi

3-sets-of-two-dozen-kites-per-type

This was created using Stella 4d, a program you can buy, or try for free, at http://www.software3d.com/Stella.php.

Five Polyhedra Featuring Kites, as Well as Other Polygons, as Faces

Posted on 3 August 2016 by RobertLovesPi

90 faces 30 rhombi and 60 kites

Convex hull of trunc icosa and dual compound

octagons and kites

kites and pentagons

kites traps and pentagons

Software credit: I used Stella 4d (available at http://www.software3d.com/Stella.php) to create these polyhedral images.

An Oblique Truncation of the Tetrahedron

Posted on 17 July 2016 by RobertLovesPi

kite-bounded tetrahedron

This polyhedron has sixteen faces: four equilateral triangles, and a dozen kites. It was created using Stella 4d, which may be found at http://www.software3d.com/Stella.php.

A Polyhedron Featuring 180 Kites as Faces, Plus Related Polyhedra

Posted on 3 October 2015 by RobertLovesPi

If one starts with the great rhombicosidodecahedron, then makes a compound of it, and its dual, and then forms the convex hull of that compound, this is the result:

This polyhedron has 180 faces, all of them kites. What’s more, there are equal numbers — sixty each — of the three different types of kites in this polyhedron.

It also has an interesting dual:

These virtual polyhedral models were created using Stella 4d: Polyhedron Navigator, which you can buy, or try for free, right here. Stella contains a “try to make faces regular” function, and here is what appears if that operation is applied to the dual shown above:

The dual of this figure is similar to the original polyhedron at the top of this post, featuring 180 kites, again: sixty each, of three different types: