The image below is a study of the geometry of MIT's Kresge Auditorium.

Projected eighth-sphere versus Reuleaux triangle

I found an article by Ivars Petersen claiming that this building's floor plan is "close to the geometry of a Reuleaux triangle" and I wanted to determine whether that was true. Other sources such as a 50-year retrospective published by MIT state that the roof of the building has the shape of an eighth of a sphere (a spherical right equilateral triangle); see this link on making a 3d model of the shape for an amusingly-captioned visualization of its construction.

So, the floor plan is the projection of an eighth-sphere; what is this shape? The edges of the roof are great circle arcs in 3d, so they project to ellipses in 2d. By my calculation, the aspect ratio of these ellipses is \( \sqrt{3}:1 \). To see this, let the sphere be the unit sphere in 3d, with the three corners of the roof at \( (1,0,0) \), \( (0,1,0) \), and \( (0,0,1) \), and project it onto the plane \( x+y+z=0 \). Then the semimajor axis of the ellipse is the radius of the sphere, \( 1 \), while the semiminor axis is the distance from the origin of the projected midpoint of an arc. The midpoint is \( \sqrt{2}(1/2,1/2,0) \), its projection is \( \sqrt{2}(1/6,1/6,-1/3) \), and the distance is \( 1/\sqrt{3} \). So I drew three ellipses with that aspect ratio, rotated by a third of a circle around their common center, and to complete the illusion of being three-dimensional (though really it's just a 2d drawing) I added another circle, with radius equal to the semimajor axis of the ellipses. Those are the grey and black parts of the figure. The shape of the auditorium floor plan is the central triangle outlined by black arcs.

The red circles in the drawing are centered at the corners of this triangle, and pass through the other two corners. Their intersection forms a Reuleaux triangle, overlaid on the other curved triangle formed by the projected roof. As you can see, the floor plan is not actually a Reuleaux triangle. It differs from Reuleaux in two significant ways: It has slightly less area, and it has elliptical arcs for sides (with variable curvature, bendier near the corners and flatter near the centers of each side) rather than circular arcs. On the other hand, as Petersen stated, it is very close.

So, to state the obvious, not all curvy triangles are alike! Another example of this same phenomenon is given by the rotor of the Wankel rotary engine: also a curved triangle with sharper angles than the Reuleaux, but with another kind of curve for its sides (the envelope of an epitrochoid). I'm pretty sure this envelope is not an ellipse, even though I don't know how to draw it. And the angles are definitely different. So the Wankel would be yet another kind of curved equilateral triangle that differs from the first two.


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