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About RobertLovesPi

I go by RobertLovesPi on-line, and am interested in many things, a large portion of which are geometrical. Welcome to my own little slice of the Internet. The viewpoints and opinions expressed on this website are my own. They should not be confused with those of my employer, nor any other organization, nor institution, of any kind.

92 Dodecahedra, Arranged as a Single Rhombic Triacontahedron

With 92 dodecahedra, if you arrange them just right, you can make a model of a rhombic triacontahedron:

RTC Augmented DodecaFor purposes of comparison, here is what the rhombic triacontahedron normally looks like:

Rhombic Triaconta

Also, referring back to the first model shown, here is a picture of just one of the red rhombi-made-of-dodecahedra.

RHOMBUS Dodeca

The first polyhedron shown in this post has an interesting dual, as well. Here it is, colored by face-type (position within the overall shape):

Dual of RTC Augmented Dodeca

Here is another view of the dual, colored by number of edges per face.

Dual of RTC Augmented Dodeca

Here’s one more view of the dual, in “rainbow color mode.”

dual RTC Augmented Dodeca rainbow

Returning to the original model, at the top of this post, here’s what it looks like, if colored by face type:

RTC Augmented Dodeca face type

Here’s one more view, in “rainbow color mode.”

RTC Augmented Dodeca rainbow

All of these images were created using Stella 4d, a program you can buy, or try for free, right here.

Building a Rhombic Enneacontahedron, Using Icosahedra and Elongated Octahedra

With four icosahedra, and four octahedra, it is possible to attach them to form this figure:

Augmented Icosa

This figure is actually a rhombus, but the gap between the two central icosahedra is so small that this is hard to see. To remedy this problem, I elongated the octahedra, thereby creating this narrow rhombus:

narrow rhombus

It is also possible to use the same collection of polyhedra to make a wider rhombus, as seen below.

wide rhombus

These aren’t just any rhombi, either, but the exact rhombi found in the polyhedron below, the rhombic enneacontahedron. It has ninety rhombi as faces: sixty wide ones, and thirty narrow ones.

REC

As a result, it is possible to use the icosahedra-and-elongated-octahedra rhombi, above, to construct a rhombic enneacontahedron made of these other two polyhedra. The next several images show it under construction (I “built” it using Stella 4d, available at this website), culminating with the complete figure.

panelnof five rhombi

panel of ten rhombi

bowl towards rec

giant rec about half complete

giant rec almost finished

giant rec complete

Lastly, I made one more image — the same completed shape, but in “rainbow color mode.”

giant rec complete rainbow

A Chiral Polyhedron with 120 Pentagonal Faces, Together with Its Dual

120 pentagons half of each type

In this chiral polyhedron, sixty faces are the small, purple pentagons, while the other sixty are the larger, orange pentagons. The next image shows its dual.

120 pentagons half of each type the dual

Both images were created with Stella 4d, a program you can buy, or try for free, at this website: http://www.software3d.com/Stella.php.

Some Enantiomorphic-Pair Compounds

In the last post here, three different color-versions of the same cluster-polyhedron were shown. Since this cluster-polyhedron is chiral, it is possible to make a compound of it, and its own enantiomer (or “mirror-image,” if you prefer). This first image shows that, with the face-color chosen by the number of sides of each face.

c5c augmented with snub cubes Compound of enantiomorphic pair

Shown next is the dual of this figure, also colored by the number of sides of each face.

c5c augmented with snub cubes Compound of enantiomorphic pair dual colored by number of sides of each face

Next, another image of the first compound shown here, but with the colors chosen by face-type (referring to each face’s position in the overall polyhedron).

c5c augmented with snub cubes Compound of enantiomorphic pair colored by face type

Finally, here is the dual, again, also with colors chosen by face-type.

c5c augmented with snub cubes Compound of enantiomorphic pair dual colored by face type

All four of these images were generated with Stella 4d, a computer program available at http://www.software3d.com/Stella.php.

The Compound of Five Cubes, Augmented with Thirty Snub Cubes: Three Versions

Cubes 5 augmented by 30 snub cubes

This cluster-polyhedron was made with Stella 4d, software you can try at this website. Above, it is colored by face-type, referring to each face’s position within the overall cluster. In the image below, the original compound of five cubes contained one cube each, of five colors, and then each snub cube “inherited” its color from the cube to which it was attached.

Cubes 5

In the next version, the colors are chosen by the number of sides of each face.

Cubes 5

The Final Stellation of the Compound of Five Icosahedra

Stellated 5 Icosahedra final stellation

This was made using Stella 4d, software available at http://www.software3d.com/Stella.php.

A Cluster of Thirty-One Rhombic Enneacontahedra

31 RECs

The rhombic enneacontahedron has thirty faces which are narrow rhombi, and sixty faces which are wider rhombi. It is also known as a vertex-based zonohedrified dodecahedron. To create this cluster-polyhedron, I started with one rhombic enneacontahedron in the center, and then augmented its thirty red faces (the narrow rhombi) with additional rhombic enneacontahedra. In the image above, I kept the yellow color for all the wide rhombi, and red for all the narrow ones. In the next image, however, the rhombi are colored by face type, referring to their position in the entire cluster-polyhedron.

31 RECs 2

Software credit: I created this using Stella 4d, software you can buy, or try for free, at this website.

Imp: A Painting from 2004

imp 2004

The medium I used for this 2004 painting, Imp, was acrylic on canvas. It turns out I have blogged this before, here, and I did not mean to create a duplicate post — but the colors appear different in the two photographs, so I have decided to leave them both on-line, anyway.

Op Art from 2010

op art from 2010

I recently found this “lost artwork” I made in 2010, two years before starting this blog. Traditionally, “op art” is black and white, only — but “traditionally” is a word seldom applied to anything I do, and I see no reason to change that now.

Rebecca West, on Feminism — and My Own, Personal Reasons for Calling Myself a Feminist

feminism

Of my two parents, one (no longer living) was a misogynistic, manipulative, abusive monster, with a list of surviving victims longer than this entire blog-post. My mother, however, is living, and has always been a feminist. I was raised by one loving, feminist parent, while constantly doing mental, verbal, and sometimes even physical battle, in self-defense, against my other parent — as a matter of survival.

This accounts, I am certain, with the fact that, to this day, it is far easier for me to form friendships with women than with men. Simply put, it is difficult for me to trust men. Men commit an overwhelming majority of the murders which happen, as well as virtually all of the rapes, and it is male politicians, as a rule, who start most — perhaps all — of the world’s far-too-numerous wars, both in the present, and the past. When one’s earliest memory is having one’s mother save one’s own life, from death by shaken baby syndrome, at age 2 ½, inflicted by one’s own father, there is no escape from lifelong psychological fallout from such a traumatic event. This is my earliest memory, and one of the causes of my PTSD, with which I will have to struggle with for the rest of my life, for this condition, unfortunately, has no cure.

When my parents (finally) divorced, around my 20th birthday, I actually went to the trouble (and expense) to legally change my last name to my mother’s maiden name, and I did this to show everyone whose side I was on — and to shed a surname which I associate, to this day, only with negative things in my life. I regret nothing about this decision. I am glad that the monster found out about this name-change, shortly after I did it, for he deserved the pain I deliberately inflicted on him by this action.

I can follow exactly half of the Biblical commandment to “Honor thy father and they mother” (Exodus 20:12), but I cannot follow the other half, for this particular monster had no honor, nor did he deserve any, now, or at any time I can remember.

I also regret nothing about the fact that my deceased parent — the monster — is no longer able to hurt anyone, since what’s left of him is, well, underground, in the literal sense of the word. I did not attend the monster’s funeral, nor was I saddened, even in the slightest, when I learned of his death. He is completely unmourned by me — and I make no apologies for any of these things.

I do not speak, nor do I write, my original last name. There are over 1400 posts on this blog, and that name appears in none of them. The reason is simple: it is not my name.

I completely agree with Rebecca West’s perfectly-reasonable definition of feminism, shown above, and, since I do subscribe to the “radical notion” that women are actually people, I see no problem whatsoever with applying the word “feminist” to myself. I’m male, after all, only as an accident of birth, and am not going to let that “coin-flip” keep me from adopting labels of my own choosing. “Feminist” is a label I wear with pride, and for highly personal reasons, as explained above. I always have been, and will remain, opposed to any efforts (such as those from the radical religious right in America) to oppress the female majority of the population. If those efforts end up destroying the Republican Party in America — which will happen, unless they reform themselves first — then Republicans will have no one to blame but themselves, and their willingness to tolerate extreme misogyny among their own ranks.