Search Results for: Zometool

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)

Rubik meets Plato in toyland: geometry toys update

Saturday, 22 June 2024 by Bruce Rawles

Geometric toys, puzzles, and amusements have been around for who-knows-how-many millennia but in the last few decades, a convergence of technologies and mathematical understanding and explorations has exploded the possibilities and offerings. While there are countless potential geometric toy concepts, we’ll focus (at least at the end of the post) on the permutations of the Platonic Solids. If one wants to have a huge range of options (including – but hardly limited to – the famous fave five Platonic Solids), my first recommendation is Zometool – the virtues of this versatile construction system this site extolls within numerous posts. There are also 5 pages of posts with geometric toys already mentioned on the GeometryCode website, including RedHen Educational Books and Toys, Fractiles, Astro-logix, and more.

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

Before we look at Platonic Solid-inspired toys, the Hoberman Sphere is a fun toy that deserves mention; it collapses down to a dense “solid” at a fraction of the expanded circumference:

… and a related (and very fun to toss up and down and watch colors change) Hoberman Switch Pitch Ball – check out the video to see how this works.

There are countless ways to “build” a cube and the SHASHIBO Shape Shifting Box transforms into over 70 shapes, emerging from … and returning to … a cube.

Platonic (Solid) Toys and Puzzles

The most ubiquitous of the 5 Platonic solids is the (hexahedron) cube and when one subdivides its six square sides into multiple (tiled, tesselated) sub-faces interesting topological possibilities arise!

The Rubik’s Cube was invented a half-century ago, on May 19, 1974. (I just missed it by a month! Happy summer solstice 2024, BTW!) When (about that same time frame) I was an undergraduate studying Electrical Engineering (Circuits and Signal Processing) at UC Davis (California) I recall two class assignments involving the Rubik’s Cube. The first (for a computer programming class, I believe using Algol) was to solve the puzzle using any arbitrary orientation of the faces; fortunately, we were given the algorithm (not Al Gore rhythm) as implementing it was plenty challenging enough! The second class assignment (in a newly launched class titled “Computer Graphics”) was to make a 3d virtual model of the cube, and print out images like the one below, along with various assigned rotations.

configurations of a 2×2×2 Rubik's cube — The configurations of a 2×2×2 Rubik’s cube reachable using only half twists form a Nauru graph. Wolfram (2022) analyzed of the 2×2×2 Rubik’s cube via a multiway graph, the first few steps of which are illustrated above.

Here are numerous 2×2, 3×3, 4×4, 5×5 + 6×6 + 7×7 toy variations of the famous (and mathematically analyzed Rubik’s Cube (dividing each square face into 4, 9, 16, 25, 36 or 49 smaller squares) that should keep spinners and fidgeters busy for quite a while. Here’s one with 486 (9 x 9 x 6) small rectangular or square facets:

“Hungarian design teacher and serious puzzler Erno Rubik assembled his first cube puzzle in 1974 and called it the Magic Cube. After a toy agent pitched the puzzle to Ideal Toy & Novelty Company, it renamed the puzzle Rubik’s Cube and began putting it in stores in 1980.” – from the Strong National Museum of Play

It’s interesting that the product description used just about every sort of related word EXCEPT dodecahedron!

Dodecahedron puzzle: GAN Megaminx Maglev — GAN Megaminx Maglev, Pentagonal Magnetic Speed Cube with 160 Magnets, 3x3x3 Fast Smooth Tension Adjustable Anti-POP STEM Toy 3D for Kids Boys Children Practice Competition, UV Coated-Grey Base (thanks to Marisa C. for sending this link)

There are many variations of these available online, including some inexpensive options.

Here is an octahedron puzzle …

… and an example of a tetrahedron puzzle …

While there aren’t any moving parts in these icosahedra, rotating the entire shape by hand provides a mesmerizing and aesthetic demonstration of refractive color mixing:

CMY Cubes The Motus (30mm) – Cyan, Magenta & Yellow Polyhedron – Subtractive Color Mixing Optical Icosahedron – Teaches Subtractive Color Mixing – Educational, Physics & Kinetic Art Desk Object

Do-it-yourselfers with some mirrored glass (and abundant care) might try making their own Kaliedoscapes, a name coined by fellow geometer, Sara Frucht who gifted me a small 3-triangular-mirror version decades ago which I have since given away before our most recent downsizing move. What’s quite fun is to see the “virtual polyhedra” that appear when the triangles are slightly truncated, or when an object such as a small rubber ball is dropped into the corner. Here’s a cool video that I’ve shared before with more examples.

Of course, Plato would remind us – while we spin, rotate, fidget, and morph these myriad manifestations – to look for the essence behind the ephemeral, the source beyond the symbol, the reality that transcends the reflected form.

Geometric art in multiple media; what have you made or found of interest?

Sunday, 28 April 2024 by Bruce Rawles

Here’s an invitation to geometric artists (and enthusiasts) to share their creativity, explorations, and discoveries.

My wife Nancy – an artist who for years has worked mostly in acrylics – has recently been making some lovely (ceramic/clay) art while taking a variety of classes since we moved to Arizona last summer and is finding lots of great tutorials for her newfound media on YouTube and elsewhere online – alerted me to this carved (presumably clay) dodecahedron by Debi Stoliar. It has 4 faces with a radial starburst pattern and 1 face with a spiral twirled starburst. Way cool!

Over the past few decades, I’ve seen lots of great geometric art, imagery, and more in numerous media, yet occasionally new media forms cross my path in delightful ways. If you search for your favorite geometric form using the search function at the top of the sidebar of this website, you’ll likely find lots of examples. I enjoy learning and discovering new and unique ways artists are transforming geometric ideas into physical and virtual art.

If you have some geometric art (2D, 3D, or otherwise) that we haven’t featured (or haven’t for quite a while) especially if you are using unconventional media, please use our contact form with a link to your art for consideration in the next monthly post on this website and/or our social media (Facebook, YouTube, Twitter a.k.a. X, Pinterest, Instagram, or perhaps another platform we haven’t explored yet! THANKS!

Here are a few dodecahedra (and related shapes) on this website:

$fractal dodecahedron$

SMAs in SMA (SMA-ll stellated dodecahedra in San Miguel de Allende)

Thursday, 5 May 2022 by Bruce Rawles

¡Feliz Cinco de Mayo! Last week, my wife Nancy and I enjoyed a long-overdue, much-needed getaway/vacation and chose San Miguel de Allende, Mexico, a place we’d heard great things about for decades from numerous sources. We very much enjoyed our immersion into the culture, cuisine, and art that this historic city is known for. In addition to world-class dining, lots of “heart decor” and so much more, I was amazed at how many 3D geometric stars were present in this lovely city, both in numerous shops and even in permanent street art! Since artsy stars, wings, and hearts are here in abundance, there are plenty of stellated polyhedra, but the predominant variety seemed to be one of my favorite shapes, the small stellated dodecahedron (one of the four Kepler–Poinsot polyhedra) which I’ve featured in numerous places on this website. I mentioned (in my broken Spanish, resurrected from high school classes a half-century ago) to a few of the shopkeepers that a foldup version that would fit in a tourist’s suitcase might afford more sales, since shipping these lovely fabrications is probably a bit expensive.

If you want to make your own (I suggest trying in cardboard first) here’s a foldup pattern from page 219 of Sacred Geometry Design Sourcebook; you’ll want to make 6 copies of this page. Join the two edges labeled 1 and also join the two edges labeled 2. If you fold together just one copy of this page, you’ll get a double pentagonal pyramid, which is another interesting shape. However, if you fold up the 2 pentagonal pyramid halves, crease the middle shared edge, and fold the other direction, you’ll have 1/6 of what you need to make this classic shape. MC Escher fans will probably recall this classic image.

Small Stellated Dodecahedron fold-up pattern from page 219 of Sacred Geometry Design Sourcebook - Universal Dimensional Patterns by Bruce Rawles — Small Stellated Dodecahedron fold-up pattern from page 219 of Sacred Geometry Design Sourcebook – Universal Dimensional Patterns by Bruce Rawles

I’ve included an assortment of photos from this trip below, including a few other familiar geometries you’ll probably recognize; enjoy! For many more photos of our San Miguel de Allende, Mexico trip this past week, visit my personal Facebook page.

Small Stellated Dodecahedra hanging from the ceiling of the gazebo in the centro park adjacent to the Parrochia church, San Miguel de Allende, Mexico — Small Stellated Dodecahedra hanging from the ceiling of the gazebo in the Centro park adjacent to the Parrochia church, San Miguel de Allende, Mexico

cluster of brass Small Stellated Dodecahedra; San Miguel de Allende, Mexico

Small Stellated Dodecahedra and many other colorful artsy decorative items in another street vendor shop; San Miguel de Allende, Mexico

Small Stellated Dodecahedron -hanging in a restaurant interior courtyard; San Miguel de Allende, Mexico

street vendor photo: metal small stellated dodecahedra and other art objects; San Miguel de Allende, Mexico

metal stellated polyhedra decorating trees in an interior courtyard; San Miguel de Allende, Mexico — metal stellated polyhedra (one of many polyhedra) decorating trees in an interior courtyard; San Miguel de Allende, Mexico

metal stellated polyhedra; San Miguel de Allende, Mexico

interior of a metal stellated polyhedron; San Miguel de Allende, Mexico

Small Stellated Dodecahedra and other geometric lamps; street vendor lamp shop; San Miguel de Allende, Mexico

large metal small stellated dodecahedron on street corner 1 block south of Parrochia church; San Miguel de Allende, Mexico — daytime photo of a large metal small stellated dodecahedron on the street corner one block south of Parrochia church, San Miguel de Allende, Mexico …

… the same large metal small stellated dodecahedron casting light on the street corner at night; San Miguel de Allende, Mexico

courtyard art: large stellated cube; San Miguel de Allende, Mexico

reflecting sphere art in hotel lobby; San Miguel de Allende, Mexico — reflecting sphere art in a hotel lobby; San Miguel de Allende, Mexico

Other examples of small stellated dodecahedra from this website:

sculpture by Gregory Hoag of I-Connect (formerly Metaforms).

New Frontiers of the Gold Country - logo

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!