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Geometric Lamps … and Coloring Pages by HYBYCOZO / COZO

Monday, 24 January 2022 by Bruce Rawles

A while ago, geometer colleagues alerted me to some remarkable geometric lamps by COZO (many using classic Platonic Solid foundations) with elaborate patterns on each face, making for a lovely immersive light experience as these patterns are projected onto walls, ceiling, and floor. Here’s an example of a Dodecahedron Lamp with an inset Flower of Life pattern in each of the 12 pentagonal faces:

Returning to the HYBYCOZO website, I discovered that they also offer nice collections of geometric coloring pages – both free and paid options. Here is an example of one of the free patterns in the free collection. The HYBYCOZO coloring book features some fine examples of tesselations, Islamic-inspired patterns, a bit of non-Euclidean geometry, and more.

These patterns are a good complement (with minimal overlap) to the wide variety of geometric patterns in Sacred Geometry Design Sourcebook (SGDS) Universal Dimensional Patterns. This timeless resource/reference book of geometric patterns has been assisting and inspiring geometric enthusiasts since 1997.

Educational Books and Toys: RedHen Update!

Tuesday, 28 December 2021 by Bruce Rawles

It’s been a while since we featured RedHen Books and Toys with their unique assortment of hard-to-find educational materials, toys, games, and books such as Bradford Hansen-Smith’s circle folding videos, books, and supplies, Zometool, Froebel gifts, and much more! For math or geometry teachers, homeschoolers, or anyone enthused about the study and practice of geometry – or wanting to propagate that enthusiasm – Red Hen has an excellent selection for kids spanning a wide age range.

To get a sense of the generous variety of available products, I would suggest clicking on the “All Items” link in the Red Hen website left sidebar, and then scrolling through a page at a time to not miss any of the unique and interesting items; the samples below are just a small handful of what you’ll find! Here are a few of the many unique items that make great gifts or educational catalysts year-round. I’m a huge fan of Zometool! Spirograph was a big positive influence in my childhood a half-century ago, so I’ll feature those two first. Enjoy!

Zome Crystallography kit

Spirograph Design Tin Set

Geometric Coloring Cards

Kaleidograph Flora

Rolligo

Mathematics with Geoshapes

Archimedes Square Stomachion puzzle

Fold a Soccer Ball

Tetrahedron Octahedron Cube poster

How to make a model of Nested Platonic Solids with Zometool

Monday, 29 November 2021 by Bruce Rawles

I recently disassembled a favorite Nested Platonic Solid model (for transport) and decided it might be helpful to take a few photos of the process and present them in reverse order, to help anyone wanting to use Zometool to build this unique geometric nest of the 5 regular polyhedra attributed to Plato. About a decade ago, I made this Infinite Recursive Animation of these Nested Platonic Solids and zoomed in for 3 full cycles of the loop to show the relationships between the five solids and some of the details that might be missed otherwise, so this post will complement that animation. Here’s a still frame from that animation that shows the 2 outermost polyhedra (the octahedron in light green and the icosahedron in dark blue):

frame from nested platonic solid animation showing octahedron-icosahedron golden-ratio vertex relationship — frame from nested platonic solid animation showing octahedron-icosahedron golden ratio vertex relationship

Here are all the Zometool struts and hubs you’ll need:

… and now we start by assembling the innermost polyhedron, the tetrahedron from these shorter green struts and white Zometool hubs:

green struts and hubs needed to make the outer octahedron: Nested Platonic Solids model using Zometool — green struts and hubs needed to make the inner tetrahedron model using Zometool

… adding the smaller blue struts and the ubiquitous white Zometool hubs to the green tetrahedron makes a blue cube (a.k.a. hexahedron); here we can see the √2 edge length proportion:

2 polyhedra; cube outermost - Nested Platonic Solids using Zometool — 2 polyhedra; cube outermost – Nested Platonic Solids using Zometool

Now adding these “roof caps” to each of the 6 square faces of the cube…

example of roof tents defining edges of dodecahedron outermost: Nested Platonic Solids model using Zometool

… we get a dodecahedron; see also page 229 in Sacred Geometry Design Sourcebook for an “exploded view” drawing in the lower-left corner of the page for additional reference.

3 polyhedra; dodecahedron outermost: Nested Platonic Solids model using Zometool

Now assemble 12 pentagonal pyramid caps …

pentagonal pyramidal caps making icosahedron: Nested Platonic Solids model using Zometool

… and attach to each of the 12 dodecahedron faces, to make it’s dual, the icosahedron:

4 polyhedra; Icosahedron outermost: Nested Platonic Solids model using Zometool

Finally, make these asymmetrical structures (with golden ratio prorportions) from the 2 larger sizes of green struts…

… and attach these to the icosahedron vertices to complete the outermost polyhedron, the octahedron:

All Five Polyhedra - Octahedron outermost: Nested Platonic Solids model using Zometool; note the golden ratio division of octahedral edges defining icosahedron vertices — All Five Polyhedra – Octahedron outermost: Nested Platonic Solids model using Zometool; note the golden ratio division of octahedral edges defining icosahedron vertices

Enjoy! :-)

Golden Mean Calipers revisited

Sunday, 29 August 2021 by Bruce Rawles

From about 4:25 – 5:19 in this video, fine art instructor Ken Campbell reveals one of his 6 keys to designing great art – using the golden mean a.k.a. golden ratio a.k.a. sacred cut a.k.a phi proportion ≈ 1.618 (or its inverse which is roughly 62% – close enough for most small art projects – in laying out design concepts.

The ratio of AB to BC in the illustration below is in golden ratio proportion. (This “Dividing a Line by the Golden Ratio” illustration is from page 156 of Sacred Geometry Design Sourcebook – Universal Dimensional Patterns. That page also shows – with simple step by step instructions – how to make a golden rectangle (a more familiar classic construction, perhaps) using only a straightedge and compass:

I was working on another project in the office when I overheard these comments in the fine art YouTube video noted above that my wife, Nancy, was watching as she brushes up (pun intended) on techniques. I reminded her that my golden ratio compass was available for her art toolkit. The one I purchased a couple of decades ago (from Now and Zen of Boulder, Colorado, (c) 1997) appears to be no longer available, and the one I featured last year seems to be currently unavailable.

However, here are a few variations of golden ratio calipers that should work well. A couple of them also have an arm that locates the midpoint of the smaller length, and in the metal one, that additional arm’s location is reversible, so it can locate the midpoint of the longer length.

If you have artwork (2D, 3D or whatever) inspired by sacred geometry, and particularly that uses the phi (golden ratio) proportion that you’d like me to consider featuring on this site, please contact me! I enjoy sharing creativity from a wide range of media and artistic styles.

Dodecahedron lamps, anyone?

Friday, 30 July 2021 by Bruce Rawles

While looking for cool images of dodecahedra – lamps in particular – I found quite a few great examples using google, including this one which has tiny blue lights (LEDs I assume?) along the interior edges.

This led (pun intended) to some other similar neat-looking geometric lamps such as this Rhombic Dodecahedron “Infinity Prism” lamp which also has 12 identical faces, each a rhombus instead of a pentagon in this case.

So, if you happen to be in the general vicinity of San Francisco, California, I have a free dodecahedron lamp with the caveat that it (obviously) needs repair since it didn’t survive a gravity-induced bout with a patio umbrella! It survived a number of moves until we downsized and didn’t have room for it in our current (Oregon) residence, but now needs a good home for a hobbyist, possibly someone with some welding skills. Here’s the content of the ad I posted on Craigslist a few days ago (first come, first serve, of course, since there’s only one of these. :-)

After many moves, and a patio umbrella falling on this one-of-a-kind lamp when in storage, this unique lamp needs repair by someone with welding tools and ability and/or a hobbyist that wants to use the precision-crafted components in some other way, such as a pattern for building a dodecahedron shape – a Platonic Solid made with 12 pentagons – in other media or some other creative ideas. Includes standard lamp sockets, swag chain electrical cord with on-off switch in the middle of the cord. The 30 metal edge pieces are 8″ long; the 12 prismatic translucent plastic pentagon pieces are a little less than 8″ on a side; the finished dodecahedron is about 18″ high.

Here are examples of similar lamps: https://www.google.com/search?q=dodecahedron+lamp … and some of the math for geometric enthusiasts:

https://www.geometrycode.com/free/platonic-solids-fold-up-patterns/#4
https://www.geometrycode.com/free/polyhedra-math-tables-for-platonic-and-archimedean-solids/

~~If you are interested in this “needs re-assembly” lamp, please contact me.~~ Never mind, it just got picked up by a fellow geometry enthusiast I just met this evening who responded to the Craigslist post. :-)