Hexagons and Octagons

hexagons and octagons

A Tessellation Featuring Multicolored, Regular Tetracontagons, as Well as Tetraconcave, Black, and Equilateral Hexatriacontagons

40

A Radial Tessellation of Regular Decagons and Bowtie Hexagons

decagon and bowtie hexagons

This tiling-pattern could be continued indefinitely, while still maintaining its five-fold radial symmetry, giving it the overall appearance of a pentagon.

An Icosagon, with Its Diagonals

icosagon

Blue-on-Blue Dodecahedron

dodecah3

This uses enlarged spheres centered on the dodecahedron’s vertices, overlapping so that they obscure the edges. Also, the faces are rendered invisible. I created it using Stella 4d, available at http://www.software3d.com/Stella.php.

A Tessellation of Regular Octadecagons and Three Types of Hexagon

tiling B

Thirty Golden Rectangles, Rotating About a Common Axis

The third image in the last post is a faceting of the icosidodecahedron. In that faceting, the faces used are equilateral triangles, star pentagons, and golden rectangles. To make these two new images, starting with that particular faceting of the icosidodecahedron, I rendered its triangles and star pentagons invisible, leaving only the thirty golden rectangles. It’s shown twice below, simply because I wanted to show it using two different coloring-schemes.

30 GOLDEN RECTANGLES

30 GOLDEN RECTANGLES RAINBOW

I would not be able to create images like this without the use of my favorite computer program, Stella 4d, written by a friend of mine who lives in Australia. You can try this program yourself, as a free trial download, at http://www.software3d.com/Stella.php.

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

49

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:

7star

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.

Falling into a Vortex

Falling into a Vortex

This Space Station for Geometricians Has, as Outer Hulls, Twelve Trapezoids, and Six Parallelograms with One Square Window / Docking Port Each

12 Trapezoids -- and six parallelograms with square windowsl

I can’t think of any good reasons for geometricans not to have their own space station, and I know what we’d do there:  we’d work on geometry (also known informally as “playing with shapes”).

My suggestion for this space station’s design was created with Stella 4d, and you may find that program (to try or guy) here:  http://www.software3d.com/Stella.php.