Articles

A Matter of Perspective

Charles SharmanJune 7, 2017

Crossbeams is an orthogonal building system. Connections happen at multiples of 90°. How is it, then, that Crossbeams can model an equilateral triangle and hexagon with internal angles of 60° and 120°?

Triangle

Hexagon

Let's analyze the triangle from a geometric perspective. Note that the triangle contains three join2s. The center of each join2 is a triangle vertex. Given any 3 points in 3D space, a plane can be drawn that intersects all three points. So, even though the points here use a third dimension, a plane can still be drawn intersecting each of them. Notice, also, that an identical beam is drawn between each pair of vertices. Thus, when the points are viewed in a plane, an identical distance separates each pair. That's an equilateral triangle.

The hexagon builds on the triangle explanation. An equilateral hexagon can be built with 6 equilateral triangles.

Hexagon made of Triangles

If we remove the center and spokes, we're left with an equilateral hexagon.

If you dislike the geometric explanation, there's another. Orient the triangle as shown below. With the x-axis right, the y-axis up, and the z-axis out of the screen, then Cartesian coordinates can be assigned to each vertex.

Triangle with Cartesian Coordinates

Now, rotate the triangle -45° about the z-axis.

Triangle Rotated about the z-axis

Finally, rotate the triange -arctan(√2) about the x-axis to put the triangle in the x-y plane.

Triangle Rotated about the x-axis

Note, the coordinates reside only in the x-y plane, and, if you remember geometry, their positions coincide with the coordinates of an equilateral triangle with side length of 2√2.

The same rigor can be applied to prove the Crossbeams hexagon is an equilateral hexagon with side length of 2√2.

*I used rotation matrices to derive the new coordinate positions after rotation. However, the rotations are simple enough, you can derive the coordinate positions purely geometrically.