Search Results for: dimensions

Kinetic Geometric Art

Thursday, 6 May 2021 by Bruce Rawles

The dimension of time coupled with the other three spatial dimensions can add tremendously to the potential of geometric art as exemplified in these kinetic sculptures. We first looked at David C. Roy’s kinetic sculptures a few years ago, and this video provides an enjoyable update on his mesmerizing moving mandalas… and other kinetic art which leverages lots of talents including geometric sensibilities.

Moving further into full 3-dimensional kinetic art, here are more lovely examples of moving geometric art by artist Ivan Black. Among many other exquisite designs in this video, if you always wanted to see the (Fibonacci Spiral) “Whirling Squares” animated in 3D, check out this outdoor, wind-driven yellow sculpture which starts about 9:17 through about 10:03 into the video; gives new vitality to the “whirling” adjective! :-)

… and in case you missed it (or would like to be wowed again), here’s a link to a prior post from this site that links to Blooms: Phi-Based Strobe Animated Sculptures that heterodyne rhythmically in dazzling fashion, benefitting from the use of stroboscopic light applied to rotating forms with golden ratio proportions.

Tetrahedron Volcanic Points on our planet

Sunday, 28 February 2021 by Bruce Rawles

On this page (Sacred Geometry Design Sourcebook – sample – (page 224) Star Tetrahedron Fold-up Pattern), I made a table with a handful of map locations (latitudes and longitudes) that seem to (mostly) have volcanic features; I recently made a google map for a web client as a demo and then made another map of some fun places we visited in New Zealand a couple of decades ago. So now I’ve made a map of these “volcanic points” that correspond to points where a tetrahedron (actually a star tetrahedron)* would intersect the sphere of the earth. (No I didn’t calculate the minor latitude change that would account for the fact that the earth is actually a spheroid – slightly flattened at the poles, but these latitudes should be fairly close.)

* If one places one vertex of a tetrahedron on the north pole, the other 3 vertices would all be at about 19.5° South Latitude; If one places one vertex of a tetrahedron on the south pole, the other 3 vertices would all be at about 19.5° North Latitude, which you can see in the map below; the longitude points vary.

If I recall correctly when we were on the big island of Hawai’i years ago, a park ranger explained that the active volcanic activity over millennia has made the entire Hawaiian Island archipelago, but the location of the active lava formation remains at the same latitude. This would seem to confirm the fixed latitude of these “fire points” as Plato might describe them.

Speaking of maps, here’s a recent interesting NYT article about the ever-popular attempts to try to convey 3D in 2D and minimize the distortions inherent in that attempt: Can This New Map Fix Our Distorted Views of the World? I’m glad I noticed this link in the article to an interesting archive article about Buckminster “Bucky” Fuller’s Dymaxion map of the earth. If you want to try living in a geodesic, visit colleague Asha’s Pacific Domes website.

Of course, higher and higher frequency flattened geodesic nets (a.k.a. fold-up patterns) give less and less distortion, but at some point, we might well ask ourselves, why lose perspective by going to fewer dimensions and less complete and inclusive information? Our tiny view of the world is necessarily hopelessly infinitesimal, but our Inner Vision can transcend the apparent limitations of space, time, and specifics. :-)

Here’s a photo of Bucky Fuller holding a cuboctahedron – an earlier map concept; the icosahedron having more faces works better. Here is what Wikipedia says about the Cuboctahedron: “Fuller applied the name “Dymaxion” to this shape, used in an early version of the Dymaxion map. He also called it the “Vector Equilibrium” because of its radial equilateral symmetry (its center-to-vertex radius equals its edge length)”

In memoriam: Keith Critchlow

Sunday, 31 May 2020 by Bruce Rawles

I recently learned of the passing of Keith Critchlow through this obituary in The Guardian. One of my first introductions – in the early 1970s – to what has become known as sacred geometry was through the books of Keith Critchlow, notably Order In Space and later, Islamic Patterns: An Analytical and Cosmological Approach. The former book has small illustrations of the tesselations (2-D tiling patterns) comprised of regular polygons. I thought these classic regular tilings all merited a full page (along with numerous archetypical variations) – so these make up a considerable portion of my first book, including pages 16-42, plus other tiling patterns inspired by these on pages 43-95, many unique to SGDS.

Here’s an example (page 27) of one of the small sketches from Prof. Critchlow’s book that I expanded and gave an entire page, in order to facilitate photocopying, coloring – for adults or “kids” of any age, arts, crafts or what have you…

You’ll note that on these 2D tesselation pages I’ve also added “unit cell” variations that show some of the possible groupings of polygon components that can make “step and repeat” (via horizontal and/or vertical translation) patterns that tile the plane in two dimensions.

I’m not sure if Order In Space was my first exposure to the 5 Platonic Solids and 13 Archimedean Solids, but this may be likely, and pages 196-214 in Sacred Geometry Design Sourcebook – Universal Dimensional Patterns were certainly inspired – in part – by Critchlow’s illustrations. You can find other useful tables in the back of SGDS that were also helped by the example of Critchlow’s generous offerings of useful patterns, ideas, and interconnections.

Elsewhere in Order In Space, he shows a matrix grid chart of the various possible permutations of nesting the 5 Platonic Solids inside each other. Again, these lovely little sketches seemed to merit full-page treatment, so they contributed to pages 224-229 in SGDS.

When I was nearing completion of SGDS in 1997, I sent a review copy to Professor Critchlow, and he sent a very thoughtful reply which included this testimonial:

“A valuable source book of geometric patterning, very useful data, a reminder of the supra human source of such mathematical forms.”

He also provided some very appropriate advice (which certainly aligned with my own experience) in advance of the initial publication, so I added this underlined sentence to page 5 of SGDS paying homage to what must be many millennia of geometers using straightedge and compass for discovery:

“I strongly encourage experimentation and particularly duplication with your own hand.”

Thanks for many years of inspiration, Keith!

Lots more to explore!!! In the process of researching archives online as well as my own prior work to find 2D tilings and tessellations, I found this excellent Wikipedia page: List of regular polytopes and compounds, which expands this topic into many fascinating connected subjects. Here’s a particularly lovely example image from the Hyperbolic star-tilings section:

Social Distancing, Space-Filling Patterns, and Non-Dual Thinking

Monday, 30 March 2020 by Bruce Rawles

In light of the current (highly appropriate) global practice of social distancing to minimize the spread of a global pandemic, the ideas and issues of space, isolation, and closeness are now in sharper focus than ever before for billions of humans on this planet. One social extreme generally considered severe punishment is “solitary confinement” where prisoners are cut off from almost all communication with others. Human examples of the other extreme might be a crowded party or bus where many of us would gasp for “breathing room” or perhaps a “dogpile” or gymnastic feat of some sort involving cramming bodies together – phone booths and VW bugs come to mind so I guess I’m dating myself with those examples. :-) Interesting how just a few short weeks can shift worldwide perception about situations involving physical proximity that we perhaps welcomed before on occasion, but now avoid – both literally and figuratively – like the plague!

Geometric Metaphors: “as close as you can get”

In the realm of geometry, we have 2-dimensional tiling patterns or tesselations where space is filled without any gaps by simple geometric archetypes. In my first book, Sacred Geometry Design Sourcebook – Universal Dimensional Patterns (SGDS), pages 16 through 84 of the 256 pages are devoted to these grids of space-filling patterns: 30 pages of regular polygon tesselations, 18 pages of square, rectangular, and circular arc tilings, 10 pages of hexagon-based or dodecagon-based tilings, and 9 pages of pentagonal tesselations, just a tiny sampling of an almost endless assortment of permutations of patterns that can fill a 2-dimensional space and not leave any gaps. Here’s an example of regular polygonal tilings from page 27 of Sacred Geometry Design Sourcebook showing a few variations (indicated by thicker lines) of “unit cell” patterns that fill space in two dimensions:

The classic work of graphic artist M. C. Escher brought some unique and elegant examples of tilings and tesselations to the world a few decades ago. Here’s a video with a few examples and related software for making tesselation patterns in 2D. There are countless examples of both periodic (uniform) and aperiodic (non-uniform, such as Penrose) tilings in nature, art, and architecture.

What about 3D Space-Filling Patterns (Tesselations)?

One of the most familiar examples in three dimensions of a space-filling pattern is a cubic lattice. Here’s an (Escher) example of a 3-D grid (think steel girders on a high-rise building). Although not space-filling per se and not 100% density, sphere packing takes on various geometric patterns, including Hexagonal Close Packing, and Cubic Close Packing. Page 239 in SGDS has a chart of radius ratios and diagrams for Coordination Numbers of 3,4,6,8 and 12 of spherical packing, as typified in ionic chemical bonding on a molecular level. Here’s an excerpt from that page:

Here are some of the less familiar 3-dimensional space-filling patterns. and related honeycomb geometries. … One of the more exotic classes of 3-D space-filling forms is Rhombic Spirallohedra; here’s a related fold-up pattern. Russell Towle introduced me to these amazing space-filling shapes almost two decades ago.

So what if “close” isn’t close enough?

In our “mind’s eye” we can visualize intersecting shapes that don’t have physical counterparts in the everyday material world around us. One example might be two identical intersecting spheres where each sphere exactly touches the other at its center; we can use ray-tracing software with opacity less than 100% (partially translucent shapes) to model how these might look (see translucent spheres intersecting in Vesica Piscis cross-section rendering below)

translucent spheres intersecting in Vesica Piscis cross-section — translucent spheres intersecting: Vesica Piscis cross-section

The 2D “shadow” (or cross-sectional slice) of the imagined 3D shape would be the famous Vesica Piscis shape. This might be analogous to intersecting (colliding) spiral galaxies where their forms interpenetrate; it might be a bit chaotic in such a galactic struggle where each “PacMan” galaxy tries to absorb the other!

Check out Russell Towle’s amazing animations – each frame of these videos being a “3D slice” through a 4D polytope or 4-dimensional polyhedron… You can see how we only get a glimpse of what these 4D shapes must be like; a reminder that our senses are limited, distorted and fallible, yet can remind us to beyond them if we’re looking for the “bigger picture.”

We’re still dealing with separate identities, separate shapes, though. The great mystical traditions, along with modern physics and non-dual metaphysics suggest that – if time and space (since both are inextricably linked in the fabric of spacetime) are an illusion – or at least a woefully incomplete and inaccurate view of reality, then even identical forms inhabiting the exact same space and time in a completely congruent fashion would still be “separate” if their identities could be distinguished from one another somehow. Recent breakthroughs in quantum computing that further validate the theory of quantum entanglement – where entangled particles are synchronized (e.g. their spin axis) instantaneously, regardless of the magnitude of the space between them – give impetus to exploring this idea.

Beyond Close

A few years ago, when my wife was about to leave that morning for an extended planned work situation in another state, while she was asleep next to me in bed, I realized that I was missing her already! How could that be possible? That “Aha!” moment further propelled my renewed exploration into the mystical study and practice of non-dual metaphysics, as I realized that she – and ultimately everyone! – lives in my mind.

Let’s do what can with zoom, Skype, web, email and whatever other technologies make sense to stay connected and support each other, especially when we appear to be more physically isolated than ever while considering that – just maybe – the imagined distance between us might just be subject to reinterpretation. This mindfulness provides all the more reason for us to be kind, compassionate and generously gracious in our assessment of each other and ourselves – as ourselves. What if the mystical Oneness alluded to down through the millennia in poetry, song, and prose – that we usually are too busy to consider – holds the effective key to not only pragmatic concerns like economic and medical infrastructure – but also to sustainable peace of mind as well? A forgiving and inclusive perspective on life would then make space (or lack thereof) a less important variable. We’re all in the same boat, fighting the same difficult battle – the silly, erroneous belief in separate interests. Sacred geometry and being mindful of clues reminding us of our interconnectedness in the world can be guideposts for the mind’s journey back to the Oneness we never left.

Ad Rate Information

For about two decades Google has consistently ranked GeometryCode.com among its top sites for sacred geometry information of all sorts. If you have a product and/or service related to sacred geometry in particular – or interconnectedness in general – please use the form below to submit your ad proposal for this advertising opportunity. We’ll carefully review your submission to ensure a good fit with our audience and the purpose of the site and respond promptly. Our current monthly (right sidebar) ad rates (US dollars) are as follows:

534px w by 534px h (square): $125

534px w by 1068px h (tall double square): $175

534px w by 1602px h (taller triple square): $225

(Other sizes available at appropriate rates.)

Note that sizes specified above are at double resolution (to ensure sharpness for retina devices such as iOS mobile screens) and will occupy 50% of the pixel dimensions noted elsewhere.