2D Geometries

Fractile Fun

Sunday, 17 July 2011 by Bruce Rawles

Fractiles are versatile geometric toys made of magnetic diamond shaped tiles.

I first learned about Fractiles and Zometool in the fall of 1997 at a Sacred Geometry conference in Boulder, Colorado.

While Zometool is primarily a 3D modeling system, Fractiles are generally used in just 2D. Accordingly, they make a great introduction to interesting ‘flat’ geometries with negligible technical background. Since the diamond-shaped tiles are arranged on a flat surface, they take up very little space, which means that they are very portable, even when assembled in any of a gazillion patterns.

What makes the combinations of these 3 diamond shaped patterns so versatile (and unique)? The angles of the rhombic shapes are all multiples of 1/14th of a circle, so one can complete a radially arranged design, and/or make amazing tiling patterns, or just play around with very aesthetically enjoyable possibilities with the permutations.

I’ve enjoyed playing — I’m just a big kid — with Fractiles for many years. I don’t even need the excuse of having younger family members visiting to bring them out and design elegant patterns with these unique magnetic tiles in 3 diamond shapes. Here are a few examples of the endless patterns that can be made in a few minutes.

Here’s a list of features from the Fractiles website:

Award-Winning Fractiles-7 received a Parents’ Choice Award for play and educational value, quality and design.
Versatile: Use these wonderful little diamond-shaped tiles to easily create seven-fold snowflakes, starbursts, spirals, bouquets of flowers, swarms of fireflies, beautiful mandalas, butterflies, spaceships, 3-D illusions, much more.
Elegant Geometry: The versatile elegant geometry allows anyone to create endless varieties of imaginative and beautiful designs. Designs range from very simple to infinitely complex. One can spend tireless hours discovering new levels of play and design.
Infinite Possibilites: Puzzle with millions of solutions — limitless combinations and diverse designs.
Wide Appeal: Fractiles Fans range from young children to seniors, from those with no math background to advanced math researchers.
Easy to Manipulate: The MAGNETIC tiles stay put – yet slide easily on their specially textured board – a helpful feature for anyone with fine motor skill problems. Magnetic Fractiles-7 is a great travel toy!
Builds Pattern Recognition Skills: Pattern recognition skills are key to learning. Lack of pattern recognition skills is linked to learning disabilities Fractiles-7 is so much fun you may not even notice how regular play increases your pattern recognition skills.
“Fractiles is so popular with my first graders, that we have to have a sign up for equal turns!” – teacher Marilyn Bowker
“My 7th and 8th graders live for free time so they can play with Fractiles.” – Sandra Bullard, The Prentice School, “Where Children with Dyslexia Learn to Learn”
Ethical: Production Fractiles is manufactured in the USA with no child labor or exploitation.

Zometool Inspiration

Saturday, 28 May 2011 by Bruce Rawles

Carlos Neumann zometool microdo — Carlos Neumann with a Zometool ‘microdo’ model

Recently we had lunch in Denver, Colorado with Carlos Neumann of Zometool, the maker of my favorite geometric modeling tool/toy. It’s really both a fabulous design and 3D visualization modeling environment AND a fun toy for any age – even those who don’t think they like geometry. He brought along a Zometool model of the “Microdo.” See the close up photograph below:

zometool microdo photo — Zometool “microdo” model

and a screen snap of it using the new Zomepad software, a VERY cool 3D modeling app that lets one rotate, modify, translate, zoom, etc. Zometool models in real time. The screen snaps just give a hint of how fun this app is!

Zomepad Microdo model — Zomepad image of Microdo model

Here’s a chrome plated (dare we say “Chrome Zome”? :-) hub. Notice that the polygons that make up the hub are equilateral triangles, regular pentagons and golden rectangles, which provide phenomenal versatility in creating Platonic and Archimedean Solids, as well as a staggering array of amazing shapes that use these foundational geometries and their 3-fold and 5-fold rotational symmetries and much more. Stay tuned for fun and decorative developments with these shiny orb-like gems! :-)

ZomeTool Chrome plated hub — Chrome plated Zometool hub

Another thing to notice about the Zomepad models is that there are background dots: red (pentagonal), yellow (triangular) and blue (golden rectangular) that are aligned behind the model as it rotates around so that you can see when you’re lined up with an axis of rotation and/or symmetry. Pretty neat!

Here are the default models that you can explore with the current pre-release version of the ZomePad app:

ZomePad model - pentagram/pentagon fractal — ZomePad model – pentagram/pentagon fractal

ZomePad Hypercube (Tesseract) model — ZomePad Hypercube (Tesseract) – 3D ‘shadow’ of a 4D polytope

ZomePad 3D Hexagonal (honey comb) grid model

Here are some Zometool models I put together years ago:

zometool stellated dodecahedron — Zometool stellated dodecahedron

The stellated dodecahedron model hung above my office cubicle at Grass Valley Group in northern California for several years; sure made a great landmark to direct folks to my desk! :-)

Here’s another of the countless topologies and geometric variations that one can make with Zometool hubs and struts. If you enjoyed Tinkertoys (or perhaps Erector sets, etc.) when you were a kid, check out Zometool. There really isn’t anything else like it. The engineering and precision design of the hub alone is an amazing story in itself that Marc Pelletier gave us a quick tour of when I was at a sacred geometry conference in Boulder, Colorado years ago, where I met lots of wonderful geometer colleagues I’m still mostly in touch with.

ZomeTool model — Zometool model (one of countless variations to explore)

Dodecahedra Origami Luminaria

Monday, 2 May 2011 by Bruce Rawles

After seeing these at a friend’s home recently, I did a google search and found a couple of nice tutorials (Dodecahedron Star Lantern Tutorial from Mom’s Crafty Space and another from WholesomePlayTime.com) for making these Dodecahedron (Pentagram) Star Lanterns; these could also be made using the Dodecahedron foldup pattern, although this design with the 5 folded over triangles on each pentagonal face highlights the inscribed pentagram (5-pointed star) on each face in an artistic way. This is a great kid’s construction project featuring sacred geometry.

Crop Circle Mandala from Jay Goldner

Monday, 2 May 2011 by Bruce Rawles

Jay Goldner, who did the superb translation of Sacred Geometry Design Sourcebook from English into German (Mandalas Der Heiligen Geometrie), has a website devoted to the amazing cereal glyphs (a.k.a. crop circles) that appear predominantly in the UK and Europe, but evidently have been around for decades all over the world.

Despite the fact that many have been hoaxed, there is also ample evidence to suggest that many are not of human origin. My wife & I walked through a formation in the UK in the summer of 1998 that showed remarkable weaving (alternating stalks); if this was hoaxed, someone went to enormous effort! Among many other unexplained discoveries, Nancy Talbott and her colleague William Levengood of BLT Research Team, Inc. have showed changes in the magnetic fields of countless plant specimens carefully selected and analyzed relative to their coordinates from a fixed reference point. There are significant magnetic changes to both the downed crop (which is noticeably different from stalks that are intentionally bent over or snapped) AND the unaffected (standing) crop.

Jay has exhibited large prints of 8 of my 12 “Martineau Solar System Series” images in various public exhibitions in Europe.

BTW, Mandalas Der Heiligen Geometrie is currently out of print by the German publisher, which we’re working on remedying. Meanwhile, I have several copies available for customers in the US, or elsewhere with adequate postage.

Here’s an example of his artistry using the geometry of crop circles as inspiration.

Jay Goldner - composite crop circle — crop circle composite image by Jay Goldner

Sacred Geometry Communities

Sunday, 13 March 2011 by Bruce Rawles

The Geometry Code – Sacred Geometry Blog – Conversations – Global Community (this is a blog-ish page from our prior site)

Videos: interviews and more

Here’s a video interview about The Seven Hermetic Laws, Sacred Geometry and more with Alan Steinfeld of New Realities and Bruce Rawles now on YouTube (originally on Emerald Energies) recorded in Sedona, Arizona at the Global Sacred Geometry Conference in Feb 2004. Thanks also to Alan Gutierrez and Angélique DeWolfe for their efforts with this interview.

Here’s a video by Dan Shaw (a long time fellow geometer) of my commentary on the Planetary Geometries: Martineau Solar System Series artwork when it was at the Mobius in Ashland, Oregon during April, 2007.

Audios: interviews and more

This is an audio interview with Jake Steele of Author’s Audio about sacred geometry, the Great Pyramid and more.

Geometric Animations

Here’s an animation I did showing how the Platonic Solids nest perfectly in an infinite recursive 3:1 size scaled loop with each iteration oriented the same – the original video has MUCH better resolution… and the looping is seamless in the QuickTime viewer! There are more details on my blog.)

Local Community (Southern Oregon)
Ashland Geometers – a.k.a. Hedronists

We have a wonderful group of kindred geometers that meet somewhat regularly in Ashland, Oregon; we call ourselves by the whimsical name “Hedronists”. We previously had links to our collaborative video/dialogues presence on MySpace, LuminosityWeb, and ning.