How Geometry Can Establish Architectural Form

Today’s post includes a small piece rejected from a larger research project which represents - much like how the best science is driven by curiosity - that even unexpected results can be valuable. The small exercise in Dynamo was initially supposed to 1) show the connection between geometry and algebra, with geometry, of course, being central to architecture, and 2) highlight how in the real world geometry and algebra often diverge from each other.

Fig.1

The starting point was to replicate the Three Square Puzzle  - as featured on Numberphile’s YouTube channel and the Trigonography blog - in REVIT using Dynamo. The puzzle asks what the sum of angles A,B,& C are (Figure 1). Intuitively, if one looks at the puzzle, it can be deduced the sum of the three angles should be 90 degrees. And indeed this is the case with over 80 such solutions cataloged in the literature (one of which is illustrated in Figure 2). However, a very curious thing happens when trying to measure the angles and sum them in the real world. Because trigonometry often includes irrational numbers, it becomes impossible to ever achieve a perfect right angle when measuring. That’s what makes it a puzzle; the solutions only exists in an idealized mathematical world.

Fig. 2

After establishing the geometry and scaling it up to apply steel framing in REVIT, I measured the angles assuming Dynamo would miss generating a perfect right angle by an arbitrarily small amount. To my great surprise, Dynamo nailed the geometry to four decimal places, the maximum precision allowed in Dynamo at this time. It’s possible that with more precision the expected results could be generated and we’d see the sum of the angles drift away from the theoretically perfect right angle. Another alternative is that Dynamo overcomes imprecision by discretizing/quantizing the output leading to nice integer solutions, kind of like Minecraft’s logical 1m x 1m x 1m block universe.

At the end of the day, these are the sorts of curious behaviours one can expect to find when experimenting with computational architecture.