Best Answer bushindo, 17 October 2012 - 05:08 PM

My take on this

Spoiler for

There are the 'same' number of points on the cube as there are on the edge. That is because we can construct a 1-to-1 mapping between any point on the cube and the corresponding point on the edge.

Let (

Similarly, we can write

Now, we can establish a 1-to-1 relationship between any point (

It is easy to see that we can reverse the process and recover a 3-D coordinate in the cube (

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There are the 'same' number of points on the cube as there are on the edge. That is because we can construct a 1-to-1 mapping between any point on the cube and the corresponding point on the edge.

Let (

**x,y,z**) be the 3-dimensional coordinate of any point in the cube (surface included) where

**x**,

**y**,

**z**= [0,1]. In decimal form, let's write

**x**as

**x**= x

_{0}.x

_{1}x

_{2}x

_{3}x

_{4}x

_{5}... where x

_{0}= 0 or 1 is the number before the decimal point, and x

_{i}= 0, 1, ..., or 9 for i > 0. (Of course, if x

_{0}= 1, then x

_{i}must all be equal to 0)

Similarly, we can write

**y**and

**z**in the same decimal form,

**y**= y

_{0}.y

_{1}y

_{2}y

_{3}y

_{4}y

_{5}...

**z**= z

_{0}.z

_{1}z

_{2}z

_{3}z

_{4}z

_{5}...

Now, we can establish a 1-to-1 relationship between any point (

**x,y,z**) on the cube to a point

**e**on line segment [0, .111 ]. To do that, we can simply interleave digits and construct

**e**from (

**x,y,z**) as follows

**e**= .x

_{0}y

_{0}z

_{0}x

_{1}y

_{1}z

_{1}x

_{2}y

_{2}z

_{2}...

It is easy to see that we can reverse the process and recover a 3-D coordinate in the cube (

**x,y,z**) given a point

**e**. Now that we have a strict 1-to-1 matching between points on a cube to the segment [0, .111], we can do a strict 1-to-1 matching between any point

**e**in [0, .111] and any point

**s**in [0, 1] using a linear function

**s**=

**e*** (1/.111 )