Category Archives: Tilings

Single Tile Aperiodicity

Although a couple years old (2010), this result is still quite a breakthrough: Socolar and Taylor have constructed a single, painted tile that when assembled correctly, can only produce an aperiodic tiling of the plane. These are some images from the paper:




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Gaspard Monge Curvature Line Vault

While principal curvature lines seem to be currently very popular, they’ve been around in architecture since 1796.

Gaspard Monge, a french geometer, proposed the above design for an ellipsoidal vault that could be constructed out of mortarless masonry units. Not only did he come up with a completely new way to build such vaults, he actually solved a much bigger problem: how to construct freeform surfaces out of discrete units (see [1], [2], and [3]).

One major characteristic of the patterns that the principal curvature lines of a surface create is that while they are stable for small disturbances of the surface, larger disturbances will cause the overall pattern to “jump” to a new one. This kind of behaviour is called a bifurcation. Now, designing something that wants to bifurcate every time it is modified is fairly difficult. Maybe an understanding of what kinds of surfaces have the same principal curvature line pattern would make this easier? Limiting the design to such surfaces would embed the rationality of the curvature line pattern directly into the design process.

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Principal Curvature Lines

Here’s a quick video of the component in action. There’s still a lot of work to be done.

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SPM Custom GH Components

The Studio for Progressive Modelling (SPM) has  started development of a couple custom GH components:

This one produces a principal curvature line emanating from a point on a surface – lines that are useful for constructing panelizations of freeform surfaces consisting entirely of planar quadrilaterals and prismatic structural members. It has been proven that surfaces with these characteristics are significantly more cost effective. However, the aesthetics of these panelization schemes are often quite pronouced, and so this component is a light weight solution that allows designers to see what kind of principal curvature mesh a given surface might suggest.

See Evolute  and Daniel Piker’s work in this area for other aspects of panelization with planar elements. I find the latter approach very interesting, as it does not depend on a setting out the curvature lines as an initial guess – even though these are fundamental to the construction. Or are they?

We’re also working on some Rhino to Revit/SAP links, and hope to post some information on those soon.

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June 10th: AiC Complexity

The Studio for Progressive Modelling would like to invite you to the third Architecture in Combination discussion.

June 10th., 5:30pm. @ 207 Queen’s Quay W., Suite 550.
Please RSVP to:
spm “at”


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Musical Sequences

Does anyone know why they’re called that? The name is so suggestive and the theory is so nice, that I can’t help but hope there’s some underlying connection, but so far, I can’t find one.

To explain, a musical sequence is an infinite list of two symbols, usually L and S (denoting “long” and “short”). In addition, this list has the property that no section of it can be copied in a regular way so as to create the full list. Another way to say the same thing is musical sequences are non-periodic. They can be generated by repeatedly sending S to L, and L to LS. So, for instance:


The incredible thing is that they seem to pop up everywhere in the world of non-periodic and aperiodic patterns (Grunbaum and Sheppard). In fact, once 3d versions of aperiodic tilings were discovered, sure enough, 3d versions of musical sequences were also found. I also haven’t found a good explanation as to why this is the case, but it seems that musical sequences create a kind of underlying organization for less organized, aperiodic, patterns.


Their role as organizers is actually what is so useful to design. Frequently, the design intent requires a facade or plan layout to be “random”. The problem is, true randomness is a pretty ugly and unwieldly thing, so more often than not, a pattern that gives the appearance of randomness is used instead. In fact, if such a pattern were to be guided by musical sequences, all sorts of pragmatic issues, like whether or not the nodes of the glazing have any relation to the nodes of the structure that holds it up, get resolved. My personal favorite 3d aperiodic tiling, developed by L. Danzer, is currently on display in a Cecil Balmond exhibition.

The point is, here is a very nice mathematical principle that will improve the constructability of certain kinds of projects while maintaining a certain aesthetic and seems to imply some connection with music. It would be perfect, except that the connection to music is still a mystery to me.

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Filed under Architecture, Exhibitions, Geometry, Tilings