Like all nulloids, this has zero volume. It may be seen as four intersecting, regular hexagons. Some consider nulloids a subset of polyhedra; others do not.
I made this using Stella 4d, which you can find at http://www.software3d.com/stella.php.
Check out http://www.software3d.com/stella.php to try the software used to make this image.
As you can see, these circles intersect on the sides of the triangles. I did not expect that, nor have I proven it. I have moved the triangle around to check to see if this remained true, and it did pass this test. If I can figure out a proof for this, I’ll post it; if one exists already, please post a comment letting me know where to find it.
Later edit: I found out that these points of intersection are the altitude feet. Here’s a diagram showing the lines containing the altitudes, meeting at the orthocenter. These blue lines also contain the angle bisectors of the brown triangle defined by the altitude feet.
If you are interested in the history of this polyhedral cluster, please see the previous two posts. Also, here’s another color scheme for it:
These images were produced using Stella 4d, software you can find at www.software3d.com/Stella.php.
This cluster was formed by putting an octahedron of the same color on each face of the compound of five tetrahedra, seen in the previous post.
In the next post, each outermost face will be augmented with an icosahedron of the same color.
This image was produced using Stella 4d, software you can find at www.software3d.com/Stella.php.
What would happen if each face in this compound were to be augmented by an octahedron of the same color? To find out, just see the next post!
I produced this image using Stella 4d, software you can find at www.software3d.com/Stella.php.
I was surfing the universe of polyhedra with the program Stella 4d (you can find it at http://www.software3d.com/Stella.php), and stumbled across this. The triskelions themselves don’t spin — although they do seem to if you watch this long enough.
In the last post, I showed you a cluster of rhombic triacontahedra (RTCs) which took a long time to build, for I had to carefully choose which faces to augment (attach) with new RTCs.
This one, by contrast, was very fast. I simply took one RTC, and augmented all thirty faces with RTCs. Unlike the RTCs in the last post, these intersect — the primary reason its appearance is so different.
In this polyhedron, many familiar polyhedra are defined. The yellow rhombi, for example, are found at the vertices of an icosidodecahedron. The blue “holes” have their centers at the vertices of an icosahedron, and the pink areas’ centers are at the vertices of a dodecahedron.
If you follow this blog closely, you have noticed that I always include this link (http://www.software3d.com/Stella.php) any time I post images made with the program I use for creating these. It’s called Stella 4d: Polyhedron Navigator. With it, you can create things of beauty, combining simplicity and complexity, which no one has ever seen before. A friend of mine on the other side of the planet wrote it, and I always include this link, where people can try or buy this software, as a way to thank him for doing so. For investigations of polyhedra, I know of no better tool.
Rhombic triacontahedra, due to their multiple symmetries, make excellent building blocks to make shapes resembling other polyhedra. This icosahedral cluster is far from being the only such possible “cluster” polyhedron made entirely of rhombic triacontahedra.
This .gif was created with Stella 4d, software you may find here: http://www.software3d.com/Stella.php.
RobertLovesPi.net 