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 Geometrical Tiling Featuring Regular Icosagons, Pentagons, and Squares; Equilateral Octagons; and Equilateral, Octaconcave Hexadecagons

tiling

The only reason I am calling this simply a geometrical tiling, rather than a tessellation, is that I want to recognize the regular icosagons (twenty-sided polygons) as part of the pattern — and the icosagons here overlap, violating the established rules for tessellations.

Hexagons and Circles

hex

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.

Two Different Sets of Two Dozen Flying Kites

24 kites

Because (6)(4) = 24 = (8)(3), that’s why.

bowtie octagon dual

I used Stella 4d to make each of these. A free trial download of this software is available here.

Zome: Strut-Length Chart and Product Review

This chart shows strut-lengths for all the Zomestruts available here (http://www.zometool.com/bulk-parts/), as well as the now-discontinued (and therefore shaded differently) B3, Y3, and R3 struts, which are still found in older Zome collections, such as my own, which has been at least 14 years in the making.

Zome

In my opinion, the best buy on the Zome website that’s under $200 is the “Hyperdo” kit, at http://www.zometool.com/the-hyperdo/, and the main page for the Zome company’s website is http://www.zometool.com/. I know of no other physical modeling system, both in mathematics and several sciences, which exceeds Zome — in either quality or usefulness. I’ve used it in the classroom, with great success, for many years.

Falling into a Vortex

Falling into a Vortex

Pentagonal System of Six Vortices

pentagonal system of vortices

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.

Two Views of a Mandala with Fivefold Symmetry

Euclidean Mandala with construction lines

Euclidean Mandala without construction lines