Bob - Penrose Tiling Generator and Explorer



Bob

Bob is a Microsoft Windows program designed to produce and explore rhombic Penrose tiling comprising two types of rhombus which together form an infinite, aperiodic plane.  In particular, Bob allows the user to discover and examine geodesic "walks" within the tiling, some of which display beautiful, complex, five-fold symmetrical patterns - "Flowers".  These Flowers appear to increase indefinitely in size and complexity as the tiling grows in extent.  

  Bob - a Windows-based Penrose Tiling generator and explorer.

Click here to download and run the software. When prompted, chose to Open or Run the installation prgram - BobI.exe.

Bob is so named after my father, Dr. Bob Collins, who discovered these walks within rhombic Penrose tiling while working as a member of the Physics Department of the University of York in England.  For a more technical description of the work, done with Tohru Ogawa of the University of Tsukuba, Japan, see the published article.  For the mathematically-minded, a technical document in Adobe PDF format describing various aspects of Penrose tiling can be downloaded by clicking here.  (It comprises scanned pages which in turn were produced by photocopying a book 20 or so years ago, so please forgive its slightly messy nature). The PDF file is also rather large (c. 9MB), so please be patient if it does not appear instantaneously!

 Dr. Bob Collins

Feel free to email me with any problems or comments with the software. 

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There is a wealth of information online about Penrose Tiling, which was discovered by the British mathematician and physicist Roger Penrose, which I do not propose to duplicate here.  Briefly, Bob deals with the type of Penrose tiling comprising two types of rhombus of equal side length.  Skinny Rhombi have interior angles of 36º and 144º; Fat Rhombi have interior angles of 72º and 108º.  These two types of rhombi can be fitted together to form an infinite, aperiodic (i.e. non-repeating) tiling.  All references to Penrose Tiling within this Web site pertain to this type of tiling.

For more information, see Penrose's book The Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics , Oxford University Press, 1989.

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Deflation is the process of producing planes of Penrose Tiling by splitting each individual rhombi of an existing plane into adjacent clusters of smaller rhombi.  All these new clusters collectively make up the next "generation" of the tiling.  The ratio of the side length of new rhombi to that of their "parent" rhombus is equal to the Golden Mean (1.618033...).

The splitting up of individual rhombi is done according to strict geometrical rules to ensure that the tiling increases indefinitely in size (as measured by the number of rhombi contained) without aberrations occurring.   In particular, each rhombus has a specific orientation, which is of no subsequent relevance to the discovery and investigation of the geodesic walks, and so is not displayed by Bob.

Bob starts off with a single rhombus - fat or skinny - as the initial seed for a tiling, and uses deflation to produce successive generations of tiling.  The number of rhombi in any given tiling is larger than its predecessor by a factor converging on (approximately) 2.6 as the number of deflations increases.

Each Fat Rhombus deflates into five smaller rhombi - three fat and two skinny. becomes
 
Each Skinny Rhombus deflates into four smaller rhombi - two skinny and two fat. becomes

Thus the side length of the rhombi in successive generations from a single seed is inversely proportional to the number of deflations required to produce the tiling.

 

As the number of deflations increases, the shape and area of the tiling converges.  Below are the perimeters of the eighth and twelfth generations produced from an initial Fat Rhombus, shown at the same scale.

Eighth Generation - 3,603 rhombi Twelfth Generation - 179,234 rhombi

Again, more details about the process of deflation, including the specific geometric rules for splitting up existing rhombi into smaller ones, can be found elsewhere on the Internet.

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Redundancy is inherent in the deflation process.  When adjacent rhombi are deflated, they will often produce a pair of child rhombi which are identical in type, position and orientation, one exactly "covering" the other.  These redundant pairs are trapped during the deflation process by Bob, and one rhombus of each redundant pair (chosen arbitrarily) is eliminated in order to guarantee a correct deflation, and thus a tiling without aberrations.

Another characteristic of deflation is that as the tiling grows, it often surrounds "holes", typically near the perimeter of the tiling, which can then persist from one deflation to the next.  These holes do not signify an error in the deflation process. 

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Now comes the interesting bit.

Dr. Collins discovered that, by following certain simple rules for traversing rhombic Penrose Tiling, a variety of different patterns or "walks" becomes apparent within the structure of the tiling.  The first step in tracing these walks is to mentally convert the rhombic tiling into a triangular tessellation by inserting an extra edge across the shorter diagonal of each rhombus.  Then chose an arbitrary vertex within the tiling as a starting point, and chose an arbitrary edge extending out from the vertex.  Follow this edge until you reach the next vertex, and then take the first edge on your left (or right).  Follow this edge until the next vertex and then take the first edge on your right (or left).  Carry on alternating between left and right to trace out the walk thus:

 

"Spiky" walks such as the closed loop displayed here can be difficult to follow visually, so for the sake of clarity, by default Bob displays walks by joining the midpoints of adjacent steps of the walk, thus:

All subsequent images of geodesic walks in this Web site use this simplified, non-spiky display mode.

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Above is an example of a walk (a Small Circle) whose path is roughly circular, and comprised of 40 steps.  A larger version (a Large Circle) contains 80 steps, an example of which is displayed below:

The perimeter of the Large Circle is more wobbly than that of the Small Circle, but it is classified as a circle nevertheless.  As yet, no circles of more than 80 steps have been discovered.  

The walk displayed below may also seem to warrant classification as a circle for the same reason as a Large Circle, but in fact its five-fold symmetry means that it is classified as a Flower (see later).

The walk displayed below is something of an oddity, in that its symmetry is unlike that of any other walk discovered within the tiling.  It has been christened a Peanut, and has 48 steps.  The right-hand image shows the distribution of Peanuts within the ninth generation tiling from an initial Fat Rhombus.

Some walks (Straight Lines) travel across the tiling without wavering from a macroscopically straight path.  Straight Lines are always oriented in one of five directions.

The final, and most intriguing classification of walk, contains those which are self-crossing.  These are called Flowers, and are by far the most complex and beautiful walks found within rhombic Penrose Tiling.  The path of a Flower consists of numerous tight loops back on itself, each loop overlapping its neighbours by no more than the width of a single rhombus.

There are two types of Flower - Stars and Pentagons - each displaying characteristic five-fold symmetry.

1,200-step Star 3,180-step Pentagon

There seems to be no limit to the size and complexity of Flowers.  The more rhombi there are in the tiling, the larger the Flowers contained within it.  Equally, no matter how large the tiling, there are always Flowers which extend tantalizingly beyond the perimeter.  The left-hand image below displays part of what appears to be a very large Star (although the self-similarity of the Flowers at different scales makes it difficult to know for sure).  This incomplete fragment of walk alone contains over 33,000 steps.  The right-hand image is a detail of bottom of the same walk, to illustrate the scale.

The self-similarity at different scales mentioned above is illustrated more clearly below.  The left-hand image shows a magnified view of the centre of the walk shown above - the shape of this small section of the walk mimics the overall shape of the walk shown above.  Similarly, taken in isolation, the three right-hand points of the partial walk in the left-hand image below are indistinguishable from the corresponding points of the 4,740-step Star in the image to its right.

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The simplest way to create a new geodesic walk within a given tiling is to first specify the starting point of the walk by double-clicking the right-hand mouse button on the targeted rhombus.  This brings up a window allowing the user to specify the direction that that the walk should take:

The legend in bold at the top of the popup window ("Click on first vertex") prompts the user to select the starting vertex of the new walk by clicking on its numbered circle covering the desired vertex.  Having selected the starting vertex, the next is selected in the same way.  Finally, the user chooses to turn either left or right at this second vertex, and the walk is then determined.

Note that, by default, the rhombus selected as the starting point of the walk is highlighted in red on the tiling underneath the popup window.  The Switch button allows the user to change the starting rhombus to any of its neighbours on the tiling.  Pressing and holding down Alt-S on the keyboard makes the selected rhombus dance around in a pleasing fashion. 

Once the walk has been determined by selecting first and second vertices and a turn direction, clicking on OK will remove the popup window and the walk will be displayed.

Walks can also be discovered automatically by Bob using the AutoWalk feature on the Walk menu.

Selecting this option displays the AutoWalk window, which is rather confusing and badly designed.  When I get a chance I'll rebuild it in a more intuitive and sensible way.  In the meantime, it can be used to automatically find walks of various types and lengths within the current tiling. 

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So, what does it all mean?  What is the significance of the symmetry and complexity of the Flowers?  Do they extend infinitely, getting more and more complex, with more and more levels of fractal-like self-similarity?

I have no idea.  Feel free to use the software and see if you can work it out for yourself, and email me with any questions or comments.  Bob can be downloaded and installed from the "use the software" link above - when prompted, chose to Open or Run the installation program, BobI.exe.

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PhiMatrix
Written by Gary Meisner: "PhiMatrix is a design and analysis application for Windows or Mac inspired by Phi, the Golden Ratio. Use it to unveil the beauty of nature's proportions or apply it to your own designs and artistic creations. It overlays a variety of tranparent grids over any other program to let you see and apply the golden ratio and other proportions to any image, and is a great tool to achieve excellence in design and composition."

Penrose Patio
Michael Fricke: "Inspired by Dan Schectman's discovery of quasicrystals, an Ohio chemist made this Penrose patio in his backyard."

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