Creative Geometry of Close Packing Spheres

I first developed the Dynamic Sphere Geometry in the late sixties; where spheres transform from one close-packing relationship to another, usually in finite sequences. The geometry intrigued me because most of the sphere arrangements generated had never been seen before and were unique. Since first presenting the Dynamic Sphere Geometry, in the early seventies, I have used the geometry to generate about one hundred arrangements – but the geometry should be capable of generating many thousands and, possibly, millions, of new and unique close packing sphere arrangements. For those interested, the whole idea started in 2D by just generating close packing circle arrangements. Circle packing was used by many cultures in the past to generate design and architectural forms: Celtic, Chinese, Islamic, Gothic…

Close-packing spheres are space efficient and have a high structural integrity, but, generally, architects, designers, and those modeling anything from molecules to new types of nano-structure, only have access to a very small range of close-packing sphere arrangements, and to their corresponding lattices. The Dynamic Sphere Geometry takes the lid off this limitation and whole families and series of new space-filling structures can be generated – each with a unique property. For those that don’t know anything about close packing spheres, imagine the centers of the spheres connected to make structural lattices, or even imagine domed buildings, or spherical structures in space, orbiting the Earth, or as nano-particle materials…

In just a few words I cannot communicate the whole geometry, or where it really gets interesting, when new close-packing, and three-dimensional, sphere clusters transform, move in space, and interconnect. Anyway, here’s an introduction by way of an animation and two sets of line drawings, that will hopefully show, just a bit, of how the geometry works. The animation clip shows four transformations (Cells One, Two, Three, and Four), extracted from a series of seven, on a two-dimensional plane.

Press this link for the animation clipSpheres1b


Cells of close packing spheres are shown where the close-packing spheres of “Cell One” transform into the close-packing spheres of “Cell Two.” These are the first two packings shown in the animation clip. Various rules are imposed on the Dynamic Sphere Geometry that govern how sequences are generated. Spheres repeat infinitely over a two dimensional plane or on multiple planes in three-dimensional space. The spheres of “Cell Two” are in whole number relationhips and two adjacent sides and the diagonal of the rectangle of “Cell Two” are 3, 4, 5 triangles.


Fig 2: SHOWS HOW “CELL TWO” TRANSFORMS INTO A 3D CLUSTER THAT REPEATS INFINITELY IN 3D SPACE. The spheres of “Cell Two,” are in whole number relationships.


Labyrinth “Keys”


In my first iBook title, “Labyrinths,” published by Apple, you’ll find a proposition that “Keys,” were used to design labyrinths of all sorts, from Neolithic to Medieval times. The “Labyrinths,” title is part of the “Geometry Through Time,” series, that I have been working on, where I explore the geometrical systems used by past cultures in architecture, art, and design.  Here is a simple animation of a “Key,” development followed by a few more advanced “Keys,” included for fun. If you are interested in the possible use of “keys,” for constructing labyrinths ranging from the Cretan labyrinth to the labyrinth in Chartres Cathedral, and more, please order the title through Apple – the link is: The title will only play on an iPad but you can order an interactive PDF directly from me.


The order in which dots and line ends, of “Keys,”  are connected will change the end labyrinth, sometimes creating a hybrid maze-labyrinth. Generally, you should start by connecting in a way similar to that of the animation – or in the reverse order. After that you might like to try any variation you can think of.


Visual Imagination and Invention

This web and blog site is primarily about geometry and about new ways to model structures in space. But it is also about imagination and the need to be flexible with the “models” we create, whether they are geometric, or not; modeling anything from an atom to the way a rain drop falls, or how the weather changes.

Visual imagination comes in many forms. Sometimes there is a type of image triggering in our minds when we look at cracks on a wall, clouds in the sky, rock formations, or the twisted bark of a tree. This type of triggering is often unconscious and can surprise, or even frighten us; when suddenly we see a figure in a tree under a moonlit sky, only to find that it is just the tree.












We often complete images in our visual imaginations with way too little information. This can happen accidentally, or when we’re trying to visually understand something. For example, a rock might appear to be a sleeping man, and the image might be really persistent in our imagination, but then we get more visual clues, and we realize that it is just a rock. We do the same sort of thing when we think we recognize someone in the street, but then find we are wrong. Sometimes, when we complete a visual image with insufficient information, the image stays with us, and we find it hard to give it up, even if it is an incorrect image. A good example of this was the misunderstanding that there were canals on Mars.








A reverse process occurs when we have a mindset for how something is, such as the Earth being a sphere, or an atom looking like a mini-solar-system. The mindsets are only models that we create, and can be very useful, but also limiting. Models of an atom range from that of Democritus, 400 B.C. to quantum models and energy profiles. Each model serves its purpose but none of them capture all of the apparent properties of an atom.





If our models are too strong, or have been overly reinforced by our teachers, then we can become blind to what might really be going on, or to new opportunities or possibilities. We could have a debate as to whether the Earth could ever have been a sphere, and under what circumstances it could find this state of mathematical perfection. It’s only when we realize that the Earth is not a sphere that we can start to understand other things that are going on.

Sometimes misunderstanding what we see leads to new ideas that might apply to something else. Other times, misunderstanding something puts us on alert, when we do not have enough visual information, and we may be right to be wary. But, on the whole, in our relatively safe world, with few predators around, we need to be careful with our imaginations; sometime let them run free, but other times direct them, or question their validity. We have no real, all inclusive, mindset for quantum physics at the moment, even though many are trying to create models to help us understand what is going on – but, on the other hand, it’s wonderful that we do not have any set models for quantum effects; because possibilities remain open. The same is true for many of established mind sets about gravity, about the internal combustion engine, about almost everything – our models can inhibit our thinking. Our visual imaginations help us survive and allow us to do wonderful new things, but sometimes we are mislead by them, and they breed prejudicial ideas. It’s just a matter of how we direct our visual imaginations. We need to be aware of the dangers of the beast!

I’ve always liked the Johnny Hart, BC, comic strip that shows an improvement to the concept of a wheel; from a square wheel to a triangular one. Conceptually the mind set is going the wrong way. Instead of removing a bump, the better solution would be to add bumps, lots of them, in fact an infinity of bumps. Eliminating bumps might have worked if conceptually one could envision moving from a square (four bump wheel), to a three bump,  two bump, one bump, and no bump wheel. Would a one bump wheel be a perfect circle (is a one bump wheel possible?), is a no-bump wheel a perfect circle??





There are many ways in which we can exercise our visual imaginations. We can step back from a mindset and try to visualize the focus of our imaginations in a wholly new way. What, for example, might land vehicles look like if we didn’t use wheels? Or how can we visualize an atom in a way that helps us better understand it’s cloud like nature, as much as it’s quantum states? What if we, for a moment, put away the idea that a plane has to look something like a plane – can it look completely different? There are also many visual illusions that we can look at that might help us experience something of the way our visual imaginations work. One such is that of Salvador Dali ( a face or people standing?).









In some small ways the designs in my Images and Altair design books can also help us exercise our visual imaginations. What can you see in a particular design? Now try and see something else? And again, and again. (From Hidden Images, “Dinosaurs,” published by Running Press.

ABJAD Designs Store Coded Messages


I have been working on my latest book, “The Arabs,” – a new title in the “Geometry Through Time,” series. I’d like to share an idea originally formed in the 1960’s, and that I have been further developing for “The Arabs”; that of the possible use of the ABJAD system to store coded messages in Islamic geometric designs.

Arabic script is primarily consonantal in form, the vowels were added later and appear as the many wavy lines and dots you see above and below the consonants. The ABJAD system provides a numeric value to each consonant, so a three consonant word, for example, can have three numbers assigned to it – one for each letter. The ABJAD system also provides a means by which numbers can be manipulated, so that the numbers corresponding to a word can be transformed to correspond to more than one word. For example, numbers can be re-arranged, or replaced by multiples of ten. The numbers, ten, five, and six, can be re-arranged as five, six, and ten, or transformed to become, sixty, fifty, and ten.

It appears that certain Islamic designs were  ABJAD encoded to communicate simple messages, give instructions, or convey ideas. It does seem that many designs were encoded deliberately, and not accidentally, with meanings supported by the environment in which the designs appear. In the Alhambra, for example, many designs consist of symmetric repeats of two and three sided rosettes that can be translated into messages of love; meanings that make sense as the Alhambra was somewhat of a “pleasure palace.” I have a prayer rug that, when numbers are counted and translated into consonants, appears to give instructions with regard to positioning hands, head, and body, to pray, whilst conjuring  images for a person to reflect upon.

To explore the idea that ABJAD  messages are encoded in Islamic designs you need two things. The first is an old arabic root dictionary; preferably published in the eighteen hundreds or early nineteen hundreds. The second is that you will need to understand the geometrical system in the architectural structure, or art and design form, that you are looking at.

An example of a possible ABJAD encoded design appears on a thirteenth or fourteenth century door to a Tekke (Tehk-keh) in Turkey. See the photograph above.


To understand the system used for the design firstly view the door panels as though they are touching each other. The complicated bit is to understand the design system itself, and this takes a knowledge of geometry and some analysis. In the case of the door, the design is based on close packing circles – where the close packing is limited to the area shown in the first line drawing. The master craftsmen then drew in the core numbers in terms of five and ten sided polygons. After that there is more complex development of five and ten sided rosettes drawn within the five and ten sided polygons. The above, three, line drawings show the first steps of the construction. There were, apparently, a number of primary design systems used to embed messages and then a methodology for applying them. To save yourself the work of exploring this idea you can buy my book, as soon as it is published!! From that point you should be able to start to to test the validity of the idea and discover for yourselves if Islamic rugs, walls, spaces, doors, art forms, etc., contain stored messages and ideas. My guess is, assuming the idea to be valid, that encoded designs started to appear from about the tenth century of the Christian era, but it’s possible that the tradition, if a tradition, started well before that. I do know that numbers were used to communicate messages long before ABJAD design applications; in knots tied in ropes, for example. If you have Islamic designs that you’d like me to analyze then please let me see them – or if you have evidence that supports the idea of ABJAD encoded designs, or undermines it, then please let me know.

The door design ends up looking like the drawing on the left below.

Applying the ABJAD system to the design, based on the primary numbers in the design, 5 and 10, and using an old arabic root dictionary we arrive at the following translations: 

A Tekke is a hall or monastery known for the dances of the Whirling Dervishes. The Whirling Dervishes are known for a dance where they rotate in a circular fashion, with their right hands facing up and their left hands facing down. They wear tall hats, on top of their heads, and they whirl like stars in the night sky. They are called to dance, and gather together to do so, meeting as arranged. They are like thirsty travelers looking for water. It’s very possible that a symbolic dance arrangement, of the whirling dervishes, followed the positions of the close packing circles that form the foundation of the door’s design.

“The Arabs,” is not just another <geometry book> but, as with all the published and planned titles, in the “Geometry Through Time” series, contains new insights into the uses, and cultural applications, of geometrical systems used in the distant past through to new systems that may be used in the future.

Summer 2012 Inventor Classes






The next Young Inventor Workshop will be on July 23rd, July 24th, and July 25th. 3 to 5pm.

Please visit the Kimball Art Center site for more information:

The first Teen Inventor Workshop will be held on July 23rd, July 24th, and July 25th. 10 to 12 am.

Please visit the Kimball Art Center for more information:


An Octagon Design System


Mathematicians working within the Islamic culture, from about 900 AD, often used geometry to communicate ideas, instructions, and even to send messages using the Abjad numerological system and the numbers of sides of polygons, rosettes, and other design forms, to encode words and meanings. Walking into the Alhambra in Granada, Spain, for example, is, in a way, like walking into a visual library of images that can be literally read. The whole subject is fascinating and involves understanding the geometrical systems used in Islamic designs as well as understanding the symmetries within the Arabic root system. In about 1972 I was talking to the author, Idries Shah, about some of my discoveries. (Shah’s book, “The Sufis,” contains lot of interesting stuff about the Abjad system).  Shah ended up using a design derived from one of the geometrical systems I had discovered, based on octagons, on the dust jacket of an edition of his book, “The Caravan of Dreams.” Shah liked the fact that the system was based on rotating eight octagons around one, creating units of nine. So, in this case, he was more interested in the symbolism of the numbers than in encoding words that could be read. I have animated the system and, if you’d like to see a part of the animation, please click on the following image:

The octagon design system creates a design grid that I have included in a number of my Images design books, for example, in Images 2, published by Running Press in Philadelphia, ISBN 978-0762439096.  Many design forms can be found within the design grids created by the system. Variations of the first example, below, appear in all sorts of design and craft forms. The second image is the image that appeared on the dust jacket of Caravan of Dreams. In the late 1960’s, and early 1970’s, I spent a number of summers in North Africa and worked with craftsmen in the Kasbahs who had lost the knowledge of the geometrical systems behind the images that they were using on everything from prayer rugs to copper plates. The design forms they were using were copies, of copies, of copies, and had consequently lost much of their geometrical value. Also, most of the craftsman had no idea of how to read or understand the symbolism, or encoded meanings, of the designs they were using. Once the craftsmen understood the systems and meanings behind the designs they were able to start to add meaning and a sort of relevance to their work that was not there before. Besides the geometrical, ABJAD, and symbolic systems, and meanings, there were also other layers of meaning added to many designs by color, by script, and with stylized images. Other layers of meaning and usage were added by the position in which the art forms appeared – where they were in relationship to others, the spaces they either defined or were part of, etc. Craftsmen following old traditions were as gifted as the greatest musicians, poets, geometricians, and artists, of antiquity.

  Creative Geometry Octagon Design