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TEE POPULAR SCIENCE MONTHLY 



Again, to repeat in words slightly different from the foregoing (Fig. 

 11) considering the a units as a points (an indefinite number), the 

 square ABCD is derived from the line AB, which for convenience sup- 

 pose to be one foot in length, by letting AB with its a points move 

 through a distance of one foot in a direction perpendicular to itself, 

 that is, perpendicular to the one dimension of AB, every point of AB 

 describes a line, and ABCD contains therefore a lines and a^ points. 



Fig. 11. 



The cube ABCD-G is derived from the square ABCD which moves 

 one foot in a direction perpendicular to its two dimensions, its a lines 

 and a^ points describing a squares and a- lines respectively. The cube 

 ABCD-G therefore contains a squares, a^ lines and a^ points. 



Similarly, the four-dimensional unit is derived from the cube, 

 ABCD-G, by letting that cube move one foot in a direction perpen- 

 dicular to each of its three dimensions, that is, in the direction of the 

 fourth dimension; its a squares, a^ lines, and a^ points describing 

 respectively a cubes, a^ squares, a^ lines. The hypercube, therefore, con- 

 tains a cubes, a^ squares, a^ lines and a* points. 



Boundaries 



Now, as to the boundaries of the units, AB has two bounding points, 

 ABCD has four, two each from the initial and the final position of the 

 moving line, ABCD-G has eight, — four each from the initial and the 

 final position of the moving square, — and the hypercube* has sixteen, — 

 eight each from the initial and the final position of the moving cube. 



Bounding Lines. — Of bounding lines, AB has one (or is itself one), 

 ABCD has 4, one each from the initial and the final position of the 

 moving line and 2 generated by the 2 bounding points of that line; 

 ABCD-G has 12, — i each from the initial and the final position of the 

 moving square and 4 generated by the 4 bounding points of that square ; 

 and the hypercube has 32, — 12 each from the initial and the final posi- 

 tion of the moving cube and 8 generated by the 8 bounding points of 

 that cube. 



Bounding Squares. — Of bounding squares, ABCD has one (itself) ; 

 ABCD-G has 6, — one each from the initial and the final position of 



* This four-dimension unit is often called the * ' tesseract. ' ' 



