Chapter 2: Truth Mining

Flattening a torus without squashing it

(Detailed version)

The standard way to embed a torus in 3 dimensions is:

(xyz) = ((a – b cos B) cos A, (a – b cos B) sin Ab sin B) (1)

where a and b are the major and minor radii of the torus, and A and B are angles that vary from 0 to 2 π radians. This is shown below left.

Torus in 3 dimensions Torus in 4 dimensions
Torus (1) in (x, y, z) coords Torus (2) in (x, y, z) coords, w shown as shading

But the torus can also be embedded in 4 dimensions, as:

(xyzw) = (a cos Aa sin Ab sin Bb cos B) (2)

This new embedding can be produced by rotating each of the original red meridians 90 degrees into the 4th spatial dimension, converting the variation in distance from the central axis to variation in the w coordinate.

Seen in (x, y, z) coordinates, the meridians start out clearly being circles (above left) and appear to be squashed into vertical lines (above right), but the w coordinate (shown as shading) gives them room to retain their shape.

Seen in (x, y, w) coordinates (with z shown as shading), the meridians start out horizontal (below left) and become vertical in the w direction (below right); in both cases the circles appear as lines, because their varying z coordinate is shown only as shading.

Torus in 3 dimensions, (x, y, w) view Torus in 4 dimensions, (x, y, w) view
Torus (1) in (x, y, w) coords, z shown as shading Torus (2) in (x, y, w) coords, z shown as shading

Why a sphere can’t be flattened

(Detailed version)

Any surface can be characterised by its Euler number: if the surface is divided into polygons (or “faces”), the Euler number χ is given by:

χ = F – E + V  

where F is the number of faces, E is the number of edges, and V is the number of vertices. For a sphere divided into 8 triangles:

F = 8  
E = 12  
V = 6  
χ = 8 – 12 + 6  
  = 2  

The Euler number remains the same even if F, E or V are changed. For example, merging two triangles with a common edge to form a square reduces F by 1, but also reduces E by 1, so χ is unchanged.

To see why the Euler number of any flat surface (with no boundary) that can be divided into a finite number of faces must be zero, consider the special case where all the faces are triangles, sharing complete edges with each other. Then every edge is shared by exactly two triangles, so:

E = F / 2  

If the surface is flat, the sum of the internal angles of all the triangles, T, must total 180 degrees times the number of triangular faces:

T = 180 F  

but this sum must also be reached by adding up the 360-degree angles around every vertex:

T = 360 V  

Equating the two gives:

V = F / 2  
χ = FE + V  
  = F – (3 F / 2) + (F / 2)  
  = 0  

Since a sphere has a Euler number of 2, it can never be made flat.

Sphere divided into 8 triangles Octahedron
A sphere divided into 8 triangles, with 360 degrees around each vertex The same sphere deformed so that each triangle is Euclidean, with angles summing to 180 degrees

Euler characteristic applet

The applet below lets you play around with different divisions of the sphere into polygons, to see how the number of faces, edges and vertices change when these features are subdivided or deleted. The initial division comes from a choice of polyhedron, but you can then make as many modifications as you like.

Drag to rotate sphere, click to select features to divide or delete.

Euler characteristic
χ = FE + V
Sum of angles
around all vertices
Sum of internal
angles of faces
if they were flat
329060221 600°20 880°

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Diaspora / Chapter 2: Truth Mining (detailed) / created Saturday, 25 October 1997 / revised Friday, 9 August 2019
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