Hexagons and Circles

Posted on 5 November 2014 by RobertLovesPi

A Star with 49 Points, to Celebrate the 49-Hour Weekend Caused by the End of Daylight Saving Time, Tomorrow, in Most of the USA

Posted on 1 November 2014 by RobertLovesPi

This is the weekend that Daylight Saving Time (or DST) ends in most parts of the USA, which means that this is the only weekend of the year, here, which lasts 49 hours, rather than the usual 48.

To celebrate this once-a-year event, I created the design above, based on the number 49. I started by making one heptagram, inscribed in a circle. The heptagram I used is one of two which exist, and is also called the {7/3} star heptagon. It looks like this:

After making one of these, I then rotated it 1/49th of a full rotation, repeatedly, until I had seven of them inscribed in the circle. Seven times seven, of course, is 49, so this created one type (many are possible) of 49-pointed star. Also, I had already extended the line segments to form lines, so that this geometrical design would extend outside the circle. Next came thickening and blackening these lines, as well as the circle, and re-coloring the red points to be black, as well.

All of this work was performed using Geometer’s Sketchpad. I then took a screenshot, moved the design to MS-Paint, and used that program to add the colors seen in the image at the top of this post.

I don’t like Daylight Saving Time, and never have, but I do enjoy the end of it, when it arrives once each year, and we get our “missing” hour returned to us — the one which was stolen from one of our weekends in the Spring.

To those who live in areas which do not observe DST, such as most of Arizona, you are fortunate — at least in this one respect. Heart attacks actually increase when DST starts each year — a fact which can be easily verified with Google. There are other problems with DST, as well. Daylight Saving Time (one of the worst ideas Benjamin Franklin ever had) should be abolished. Everywhere.

A Halloween Rhombicosidodecahedron

Posted on 31 October 2014 by RobertLovesPi

This Jack-o-Lantern picture was found with a Google image-search, and then I projected it onto a rhombicosidodecahedron, and created this rotating .gif file, using Stella 4d — a program available at http://www.software3d.com/Stella.php. Happy Halloween!

A Rhombic Dodecahedron, Decorated with Curvy Tessellations

Posted on 30 October 2014 by RobertLovesPi

The faces of this polyhedron are decorated with the same type of curvy tessellation seen in the last post here, and it was created using Geometer’s Sketchpad and MS-Paint. Projecting these images onto the faces of this rhombic dodecahedron, in different colors, and then creating this rotating images of it, required a third program, Stella 4d: Polyhedron Navigator. This latter program, an indispensable tool for polyhedral investiagations, may be tried for free, as a trial version, at http://www.software3d.com/Stella.php.

A Dozen Triangula

Posted on 25 October 2014 by RobertLovesPi

This dodecahedron is adorned with images of the Triangulum Galaxy. The plural of “Triangulum” is “Triangula,” is it not?

Software credit: this rotating image was created using Stella 4d: Polyhedron Navigator, which is available at http://www.software3d.com/Stella.php.

Five Mandalas

Posted on 11 October 2014 by RobertLovesPi

A Polyhedral Journey, Beginning with a Near-Miss Johnson Solid Featuring Enneagons

Posted on 9 October 2014 by RobertLovesPi

When Norman Johnson first found, and named, all the Johnson solids in the latter 1960s, he came across a number of “near-misses” — polyhedra which are almost Johnson solids. If you aren’t familiar with the Johnson solids, you can find a definition of them here. The “near-miss” which is most well-known features regular enneagons (nine-sided polygons):

This is the dual of the above polyhedron:

As with all polyhedra and their duals, a compound can be made of these two polyhedra, and here it is:

Finding this polyhedron interesting, I proceeded to use Stella 4d (polyhedron-manipulation software, available at http://www.software3d.com/Stella.php) to make its convex hull.

Here, then, is the dual of this convex hull:

Stella 4d has a “try to make faces regular” function, and I next used it on the polyhedron immediately above. If this function cannot work, though — because making the faces regular is mathematically impossible — one sometimes gets completely unexpected, and interesting, results. Such was the case here.

Next, I found the dual of this latest polyhedron.

The above polyhedron’s “wrinkled” appearance completely surprised me. The next thing I did to change it, once more, was to create this wrinkled polyhedron’s convex hull. A convex hull of a non-convex polyhedron is simply the smallest convex polyhedron which can contain the non-convex polyhedron, and this process often has interesting results.

Next, I created this latest polyhedron’s dual:

I then attempted “try to make faces regular” again, and, once more, had unexpected and interesting results:

The next step was to take the convex hull of this latest polyhedron. In the result, below, all of the faces are kites — two sets of twenty-four each.

I next stellated this kite-faced polyhedron 33 times, looking for an interesting result, and found this:

This looked like a compound to me, so I told Stella 4d to color it as a compound, if possible, and, sure enough, it worked.

The components of this compound looked like triakis tetrahedra to me. The triakis tetrahedron, shown below, is the dual of the truncated tetrahedron. However, I checked the angle measurement of a face, and the components of the above compound-dual are only close, but not quite, to being the same as the true triakis tetrahedron, which is shown below.

This seemed like a logical place to end my latest journey through the world of polyhedra, so I did.

An Octahedron Made of Cubes

Posted on 6 October 2014 by RobertLovesPi

Software credit: I made this using Stella 4d, which is available at http://www.software3d.com/Stella.php.

A Polyhedral Journey, Beginning with Face-Based Zonohedrification of an Icosahedron

Posted on 28 September 2014 by RobertLovesPi

To begin this, I took an icosahedron, and made a zonish polyhedron with it, with the new faces based on the zones of the existing faces. Here’s the result.

Next, I started stellating the polyhedron above. At the sixth stellation, I found this. It’s a true zonohedron, and the first polyhedron shown here is merely “zonish,” because one has triangles, and the other does not. (One of the requirements for a polyhedron to be a zonohedron is that all its faces must have an even number of sides.)

After that, I kept stellating, finding this as the 18th stellation of the first polyhedron shown here.

With this polyhedron, I then made its convex hull.

At this point, the irregular hexagons were bothering me, so I used Stella 4d‘s “try to make faces regular” option. (Stella 4d is polyhedron-manipulation software you can try for free, or purchase, right here.)

The next step I chose was to augment all the yellow trapezoids with prisms, each with a height 1.6 times the trapezoids average edge length.

The next step was, again, to make the convex hull.

At this point, I tried “try to make faces regular” again, and was pleased with the result. The green rectangles became so thin, however, that I had to stop displaying the edges and vertices, in order for then to be seen.

Next, I augmented both the blue faces (decagons) and the yellow faces (dodecagons) with antiprisms, again using a height 1.6 times that of the augmented faces’ average edge-lengths.

Next, I made the convex hull again — a step I often take immediately after augmenting a polyhedron.

This one surprised me, as it is more complicated than I expected. To clean things up a bit, I augmented only the trapezoids (dark pink) with prisms, and dodecagons (green) with antiprisms, again using the factor 1.6 for the augmentation-height.

The next step I chose was to take the convex hull, once more. I had not yet noticed that the greater height of the trapezoidal prisms would cause the dodecagonal antiprisms to be “lost” by this step, though.

Next, “try to make faces regular” was used again.

This last result had me feeling my polyhedral journey was going in circles, so I tried augmentation again, but in a different way. I augmented this polyhedron, using prisms, on only the red trapezoids (height factor, 1.6 again) and the blue rectangles (new height factor, 2.3 times average edge length).

After that, it was time to make another convex hull — and that showed me that I had, indeed, taken a new path.

I found the most interesting faces of this polyhedron to be the long, isosceles trapezoids, so I augmented them with prisms, ignoring the other faces, using the new height-factor of 2.3 times average edge length this time.

Of course, I wanted to see the convex hull of this. Who wouldn’t?

I then started to stellate this figure, choosing the 14th stellation as a good place to stop, and making the edges and vertices visible once more.

A Zonish Icosahedron, and Some of Its “Relatives”

Posted on 27 September 2014 by RobertLovesPi

To begin this, I used Stella 4d (available here) to create a zonish polyhedron from the icosahedron, by adding zones along the x-, y-, and z-axes. The result has less symmetry than the original, but it is symmetry of a type I find particularly interesting.

After making that figure, I began stellating it, and found a number of interesting polyhedra in this polyhedron’s stellation-series. This is the second such stellation:

This is the 18th stellation:

The next one, the 20th stellation, is simply a distorted version of the Platonic dodecahedron.

This one is the 22nd stellation:

This is the 30th stellation:

The next really interesting stellation I found was the 69th:

At this point, I returned to the original polyhedron at the top of this post, and examined its dual. It has 24 faces, all of which are quadrilaterals.

This is the third stellation of this dual — and another distorted Platonic dodecahedron.

This is the dual’s 7th stellation:

And this one is the dual’s 18th stellation:

At this point, I took the convex hull of this 18th stellation of the original polyhedron’s dual, and here’s what appeared:

Here is this convex hull’s dual:

Stella 4d, the program I use to make these (available here), has a built-in “try to make faces regular” function. When possible, it works quite well, but making the faces of a polyhedron regular, or even close to regular, is not always possible. I tried it on the polyhedron immediately above, and obtained this interesting result:

While interesting, this also struck me as a dead end, so I returned to the red-and-yellow convex hull which is the third image above, from right here, and started stellating it. At the 19th stellation of this convex hull, I found this: