A Philosophical Investigation
In the Wittgensteinian Tradition
Once upon a time, a seafaring tribe, the Trims, were into fishing, sailing, and weaving nets.
They had a specific way of multiplying, passed down from one generation to the next.
“Three by five” would mean the surface area between two edges of length three and five, with a sixty degree angle between them. The resulting area: fifteen triangular units.
Or take “six by two” as shown below. The units of area are the small triangles at any corner. They tile the entire surface.
The green triangle consists of an area equal to twelve of these units. The two edges emanate from the lower right corner to define this “two by six” or “six by two” green triangle.
“Two by two by five” would describe three edges of a volume, again with sixty degree angles between them, each emanating from the same corner of a regular tetrahedron.
The resulting volume is two times two times five, or twenty tetrahedral units.
The unit volumes are visible at the tips. These tetrahedrons alternate with octahedrons of volume four to fill the larger tetrahedron completely.
Any area or volume, of any shape, could be related to these unit edge lengths and corresponding methods for multiplying.
Then the HMS Orthodoxy arrived, full of colonists from a Land of Right Angles. These were the so-called Ortho-normals, or Normals for short.
The Normals opened up the angles in question quite a bit wider, from sixty to a full ninety degrees.
When they multiplied two or three numbers together, they would not simply “close the lid” by connecting the edge tips. They would build a corresponding rectangle, or a rectangular prism, adding parallel members, and call these the area and volume respectively.
Here’s a summary of the two ways of reckoning:
Given the Trims had a tetrahedral unit of volume, their canonical set of polyhedrons was pretty easy to memorize. The tetrahedron plays well with others.
The cube of two intersecting unit tetrahedrons had volume three by Trim reckoning.
The cube’s dual, an octahedron (red) of edges the same length as the cube’s face diagonals, had a volume of precisely four tetravolumes.
The rhombic dodecahedron (blue), each a containment cell for a unit radius sphere, in a space-filling honeycomb arrangement (CCP), had a tetravolume of six.
Twelve spheres around a nuclear one gave a cuboctahedron (yellow) of volume twenty, once sphere centers were chord connected. This volume twenty could be twist-contracted into an icosahedron, bridging to shapes with five-fold symmetry. The icosahedron volume: 5 √2 Φ² tetravolumes.
Given volumes one and four packed together to define the vertices for the unit radius balls, the entire geometry packed neatly into a tight and memorable curriculum.
The Ortho-normals were not appreciative however, and forbade any further teaching of the Trim system.
However, in the 1900s, a philosopher would come along and rediscover the Trims’ system and try to popularize it.
The Normals still said “no way!” but their grip on power had faded in the interim, and some of the old Trim knowledge was seeping through, including right here on Medium.
Want a fast way to compute the volume of any tetrahedron, regular or not, in tetravolumes? Just input the six edges, starting from any apex and then going around the opposite base.
Using the above, we’re able to use plane nets directly, for inputs. We have the A & B volumes, each of volume one twenty fourth. We also have the S, T & E modules. Check it out!
For further reading, see Wikipedia.
