image I painted this in 2002, using acrylic on canvas.

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I painted this in 2001, using acrylic on canvas.

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This is a painting from 2002, which I did with acrylic, on canvas. I just had the first chance in years to get a good photograph of it, for it’s at my mother’s house, over three hours from where I live, and I actually have a decent camera with me for this visit.

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Can Defenders of the Police in Ferguson, Missouri Explain These Numbers?

fergusonThe source of this image is an official website of Missouri’s state government: http://ago.mo.gov/VehicleStops/2013/reports/161.pdf.

As shown above, when white residents of Ferguson, Missouri are stopped by the police, there is a higher contraband hit rate than is the case with Black residents. However, Blacks there have traffic-stop rates, search rates, and arrest rates far higher than those of whites.

Blacks in Ferguson are 63% of the population. In 2013, Blacks were stopped by the police there 4,632 times, compared to only 686 times for white drivers.

If anyone wants to convince me that the Ferguson Police Department is not a racist organization, operating, as a group, to continue America’s long history of oppression by skin color, they’ll need to explain these numbers first.

DWB (“Driving While Black”) should never be a cause for a traffic stop, but it still is, all over the USA. If you don’t believe me, conduct this simple test:  ask a Black person, old enough to drive, what a “DWB” is, and then ask if it really happens, in America, in 2014.

It would be going too far to state that all police officers are racist criminals. The fact is that many police officers do not fit that description at all. However, it is also true that many other police officers are criminals of this type, and they tarnish the reputation of all police officers, and police departments, by their actions. America should do something, now, about our “criminal police” problem. It isn’t limited just to Ferguson, nor only to Missouri.

[Credit:  Thank you, to the Tumblr-bloggers at http://sassygayklavierspieler.tumblr.com/ and http://fishingboatproceeds.tumblr.com/, for bringing this chart to my attention.]

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A Rhombic Triacontahedron, Made of Icosidodecahedra

It turns out that it is possible to use multiple icosidodecahedra as building blocks to build that polyhedron’s dual, the rhombic triacontahedron. This construction appears below, in four different coloring-schemes.

Augmented IcosidodecaAugmented Icosidodeca2Augmented Icosidodeca3Augmented Icosidodeca4

These rotating virtual models were constructed using Stella 4d, a program available at http://www.software3d.com/Stella.php.

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A Rhombicosidodecahedron, Made of Rhombicosidodecahedra

This “metarhombicosidodecahedron” took a long time to build, using Stella 4d, which you can find at http://www.software3d.com/Stella.php — so, when I finished it, I made five different versions of it, by altering the coloring settings. I hope you like it.

Augmented Rhombicosidodeca

Augmented Rhombicosidodeca2

Augmented Rhombicosidodeca4

Augmented Rhombicosidodeca5

Augmented Rhombicosidodeca6

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A Collection of Unusual Polyhedra

In the post directly before this one, the third image was an icosahedral cluster of icosahedra. Curious about what its convex hull would look like, I made it, and thereby saw the first polyhedron I have encountered which has 68 triangular faces.

68 triangles Convex hull

Still curious, I next examined this polyhedron’s dual. The result was an unusual 36-faced polyhedron, with a dozen irregular heptagons, and two different sets of a dozen irregular pentagons.

dual of 68 triangles Convex hull -- this dual has 36 faces including 12 heptagons and 12 each of two types of pentagon

Stella 4d, which is available at http://www.software3d.com/Stella.php, has a “try to make faces regular” function, and I tried to use it on this 36-faced polyhedron. When making the faces regular is not possible, as was the case this time, it sometimes produce surprising results — and this turned out to be one of these times.

dual of the 68-triangle polyhedron after 'try to make faces regular' used

The next thing I did was to examine the dual of this latest polyhedron. The result, a cluster of tetrahedra and triangles, was completely unexpected.

dual of the dual of the 68-triangle polyhedron after 'try to make faces regular' used

The next alteration I performed was to create the convex hull of this cluster of triangles and tetrahedra.

Convex hull of that triangular mess

Having seen that, I wanted to see its dual, so I made it. It turned out to have a dozen faces which are kites, plus another dozen which are irregaular pentagons.

dual of the Convex hull of that triangular mess 12 kites and 12 irregular pentagons

Next, I tried the “try to make faces regular” function again — and, once more, was surprised by the result.

dozen kites and dozen pentagons after 'try to make faces regular' used

Out of curiosity, I then created this latest polyhedron’s convex hull. It turned out to have four faces which are equilateral triangles, a dozen other faces which are isosceles triangles, and a dozen faces which are irregular pentagons.

Convex hull Z

Next, I created the dual of this polyhedron, and it turns out to have faces which, while not identical, can be described the same way:  four equilateral triangles, a dozen other isosceles triangles, and a dozen irregular pentagons — again. To find such similarity between a polyhedron and its dual is quite uncommon.

dual of Convex hull Z

I next attempted the “try to make faces regular” function, once more. Stella 4d, this time, was able to make the pentagons regular, and the triangles which were already regular stayed that way, as well. However, to accomplish this, the twelve other isosceles triangles not only changed shape a bit, but also shifted their orientation inward, making the overall result a non-convex polyhedron.


Having a non-convex polyhedron on my hands, the next step was obvious:  create its convex hull. One more, I saw a polyhedron with faces which were four equilateral triangles, a dozen other isosceles triangles, and a dozen regular pentagons.

Convex hull

I then created the dual of this polyhedron, and, again, found myself looking at a polyhedron with, as faces, a dozen irregaular pentagons, a dozen identical isosceles triangles, and four regular triangles. However, the arrangement of these faces was noticeably different than before.

latest Convex hull

Given this difference in face-arrangement, I decided, once more, to use the “try to make faces regular” function of Stella 4d. The results were, as before, unexpected.


Next, I created this latest polyhedron’s dual.


At no point in this particular “polyhedral journey,” as I call them, had I used stellation — so I decided to make that my next step. After stellating this last polyhedron 109 times, I found this:

109 stellationsTTMFRA dual

I then created the dual of this polyhedron. The result, unexpectedly, had a cuboctahedral appearance.

Faceted Dual

A single stellation of this latest polyhedron radically altered its appearance.

stellation Faceted Dual

My next step was to create the dual of this polyhedron.

dual Faceted Stellated Poly

This seemed like a good place to stop, and so I did.

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