OK, Lars and DcadRob have been musing about things mathematical, so here is a really mind-numbing problem. And there is a single simple answer.

1. First imagine a series of geometrical objects; point, line (segment), square and cube.

A point is zero dimensional - no length, width or height.

A line segment is the result of moving a point X units in a single direction - one dimensional.

The square is the result of moving the line segment X units in a direction perpendicular to the line - two dimensions.

A cube is the result of moving a square X units in a direction perpendicular to the plane - three dimensions.

A tessarect is the result of moving a cube X units in a direction perpendicular to all surfaces of the cube - four dimensional (proof left to the student, or read "And He Built A Crooked House" by Robert Heinlein).

etc.

2. Each object is bounded by the figure of the previous dimension. Cubes are bounded by squares, squares are bounded by line segments, line segments are bounded by points, etc. All are joined at right angles, making these "right regular objects."

3. Let D be the number of dimensions in an object, and N be the number of dimensions in a part of the object. For a three dimensional cube the bounding figure (square) is two dimensional, so D=3 and N=2. You could also consider the lines in a cube where N=1 and the points where N=0.

4. This table describes the number of N dimensional objects in D dimensional objects:

D \ N 0 1 2 3 4 5

0 1 - - - - -

1 2 1 - - - -

2 4 4 1 - - -

3 8 12 6 1 - -

4 16 32 24 8 1 -

5 ? ? ? ? ? 1

For example a cube (D=3) has 8 points (N=0), 12 lines (N=1), 6 squares (N=2) and 1 cube (N=3).

5. Find a single equation that can be used to calculate all relationships of these right regular objects and fill in the blanks for the pentacube (D=5). Don't cheat - figure it out for yourselves. I did. Besides, unless you know the particular branch of mathematical obscura this falls under you will probably never find it, not even with Google.

It took me almost 30 years of searching before I discovered that I wasn't the first to come up with the formula. #^*&%! It was published in 1946! So I know how you feel, Lars.

****

PS: Did you know that an 8-dimensional right regular object has 1792 lines

PPS: OK, so I exaggerated. It really isn't the universal law of Life, the Universe and Everything. But it's about as close as we will ever get.

Phil

1. First imagine a series of geometrical objects; point, line (segment), square and cube.

A point is zero dimensional - no length, width or height.

A line segment is the result of moving a point X units in a single direction - one dimensional.

The square is the result of moving the line segment X units in a direction perpendicular to the line - two dimensions.

A cube is the result of moving a square X units in a direction perpendicular to the plane - three dimensions.

A tessarect is the result of moving a cube X units in a direction perpendicular to all surfaces of the cube - four dimensional (proof left to the student, or read "And He Built A Crooked House" by Robert Heinlein).

etc.

2. Each object is bounded by the figure of the previous dimension. Cubes are bounded by squares, squares are bounded by line segments, line segments are bounded by points, etc. All are joined at right angles, making these "right regular objects."

3. Let D be the number of dimensions in an object, and N be the number of dimensions in a part of the object. For a three dimensional cube the bounding figure (square) is two dimensional, so D=3 and N=2. You could also consider the lines in a cube where N=1 and the points where N=0.

4. This table describes the number of N dimensional objects in D dimensional objects:

D \ N 0 1 2 3 4 5

0 1 - - - - -

1 2 1 - - - -

2 4 4 1 - - -

3 8 12 6 1 - -

4 16 32 24 8 1 -

5 ? ? ? ? ? 1

For example a cube (D=3) has 8 points (N=0), 12 lines (N=1), 6 squares (N=2) and 1 cube (N=3).

5. Find a single equation that can be used to calculate all relationships of these right regular objects and fill in the blanks for the pentacube (D=5). Don't cheat - figure it out for yourselves. I did. Besides, unless you know the particular branch of mathematical obscura this falls under you will probably never find it, not even with Google.

It took me almost 30 years of searching before I discovered that I wasn't the first to come up with the formula. #^*&%! It was published in 1946! So I know how you feel, Lars.

****

PS: Did you know that an 8-dimensional right regular object has 1792 lines

*and*1792 cubes?PPS: OK, so I exaggerated. It really isn't the universal law of Life, the Universe and Everything. But it's about as close as we will ever get.

Phil