Monthly Archives: October 2013

165-Srix

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165-Srix

In three-dimensional space, there are five Platonic and thirteen Archimedean polyhedra, plus numerous other shapes, in several categories. The whole collection can appear to be quite a confusing jumble — until, and unless, you start surveying four-dimensional polytopes, known as polychora.

There are six regular polychora, and they are analogous to the five Platonic solids. Each three-dimensional cell is regular, and all are identical, within a single one of these six. When peering beyond these six, however, things can get very confusing, very quickly.

The software I used to generate this image, Stella 4d, has a built-in library of polyhedra and polychora. You can examine it as a free trial download at http://www.software3d.com/stella.php. Today, motivated by curiosity, I went surveying, using this program, into the more complex polychora — beyond the six regular ones — which have different polyhedra as cells, looking for one I could (try to) understand, and which appealed to me aesthetically.

The one I settled on for this post is known as 165-Srix, as well as the small rhombated 600-cell, a/k/a the cantellated 600-cell. It has 600 cells which are cuboctahedra, shown here in yellow, 120 more which are icosidodecahedra, shown here in blue, and 720 cells which are regular pentagonal prisms.

I must admit this: I’m more than a little jealous of those who seem to be able to easily understand these four-dimensional shapes. I am definitely not one of them.

120 Undulating Dodecahedra

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120 Undulating Dodecahedra

This is a 120-cell, one of the regular polychora (four-dimensional polytopes), with its edges and vertices rendered invisible, and its dodecahedral cells shrunk somewhat, to put some empty space between them. It’s rotating in hyperspace, and what you are seeing at any given moment is a particular three-dimensional “shadow,” or projection, of the entire figure.

It’s easy to make this sort of thing with software called Stella 4d, written by an Australian friend of mine. Here’s a link to a site where you can try it, as a free trial download, before deciding whether or not to purchase the fully-functioning version: http://www.software3d.com/stella.php.

600 Undulating Tetrahedra

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600 Undulating Tetrahedra

This is a 600-cell, one of the regular polychora (four-dimensional polytopes), with its edges and vertices rendered invisible, and its cells shrunk so that they do not touch. It’s rotating in hyperspace, and what you are seeing at any given moment is a particular three-dimensional “shadow,” or projection, of the entire figure.

It’s easy to make this sort of thing with software called Stella 4d, written by an Australian friend of mine. Here’s a link to a site where you can try it, as a free trial download, before deciding whether or not to purchase the fully-functioning version: http://www.software3d.com/Stella.php.

A Ball of Kites

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A Ball of Kites

“All Is Number”

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“All is number” was a motto of the ancient Pythagorean Society. I thought it would also make a good title for this geometrical design.

When Must the Four Perpendicular Bisectors of the Sides of a Quadrilateral Be Concurrent?

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When Must the Four Perpendicular Bisectors of the Sides of a Quadrilateral Be Concurrent?

The perpendicular bisectors of the three sides of any triangle must be concurrent, as is well-known. However, this is not true of quadrilaterals. For quadrilaterals, the four perpendicular bisectors may be concurrent, or not. So when must they be?

The answer: when a pair of opposite angles in the quadrilateral are right angles.

Why is this the case? Well, it’s a consequence of what happens in triangles to this point of concurrence, called the circumcenter. In right triangles — and only in right triangles — the circumcenter falls on a side of the triangle, and that side is always the hypotenuse, with the circumcenter located at its midpoint. If a quadrilateral has two right angles as a pair of opposite angles, as ABCD does in the diagram above, then it can be split into two right triangles with a common hypotenuse, as shown — and that hypotenuse’s midpoint will then be the point of concurrence of all four perpendicular bisectors of the sides of the quadrilateral.

[Later edit:  my friend Andrew make the following comment, when I posted a link to this post on Facebook. I appreciate it when my friends make such corrections.]

“Actually, the Perpendicular Bisectors are concurrent for any Cyclical Quadrilateral. (Opposite angles sum to 180 degrees). Even Non-convex Cyclical Quadrilaterals have this property (Note a Non-convex Cyclical Quadrilateral must be self-intersecting). All Cyclical Quadrilaterals can be circumscribed by a circle.”

For proof, consider the cyclical or non-cyclical quadrilateral (or higher polygon, as well), together with its circumscribed circle. All sides of this polygon are chords of this circle, and perpendicular bisectors of chords pass through the circle’s center — the point of concurrency.

Polyhedron Featuring Decagons and Triangles

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Polyhedron Featuring Decagons and Triangles

This was created using Stella 4d, which you may try for free at http://www.software3d.com/stella.php.

A Survey of Right Interior Angles in Hexagons

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A Survey of Right Interior Angles in Hexagons

A regular hexagon, of course, has no right angles, but irregular, convex hexagons can have one, two, or three right angles.

With one right angle, there is only one basic configuration, but, with two right angles, there are three: the right angles may be consecutive, have one non-right angle between them, or be opposite angles.

There are also three possible configurations with three right angles: the three angles can be consecutive, or two can be consecutive with one non-right angle separating the other right angle from the consecutive pair, or every other angle can be a right angle.

Four right angles cannot exist in a convex hexagon, nor can five, nor, of course, six. Four right interior angles are possible, however, for non-convex hexagons, and, again, there are three possible configurations. In the first, the four right angles are consecutive. In the second, three are consecutive, then a non-right angle separates the fourth right angle from the other three. In the third, there are two pairs of consecutive right angles, with single non-right angles separating the pairs on opposite sides of the hexagon.

A Survey of Right Angles in Convex Pentagons

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A Survey of Right Angles in Pentagons

A regular pentagon, of course, has no right angles, but irregular pentagons can have one, two, or three (but not four, nor five). There are two varieties for both two and three right angles in pentagons — the right angles can be consecutive, or non-consecutive.

Compound of the Rhombic Triacontahedron and a Truncated Icosahedron

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Compound of the Rhombic Triacontahedron and a Truncated Icosahedron

I stumbled across this while manipulating polyhedra with Stella 4d, which you can try for free at http://www.software3d.com/stella.php.

The title describes the blue and yellow figure as “a” truncated icosahedron, rather than “the” truncated icosahedron, because of the slight irregularity of the hexagonal faces, a result of the truncation-planes being slightly closer to the center than is the case for the true Archimedean solid. It should be possible to fix this, but that may be beyond my abilities.