Hyper-space

Excerpts

• Hyper-space

  Space, time, and matter cannot be defined; but the means of measuring them and the investigation of their properties fall within the domain of mathematics.

• Hyper-space

  In the hyperbolic system there are no similar figures of unequal size; the area of a triangle can be deduced from the sum of its angles, which is always less than two right angles; and there is a finite maximum to the area of a triangle.

• Hyper-space

  If an inhabitant of flatland was able to move in three dimensions, he would be credited with supernatural powers by those who were unable so to move; for he could appear or disappear at will, could (so far as they could tell) create matter or destroy it, and would be free from so many constraints to which the other inhabitants were subject that his actions would be inexplicable by them.

• Hyper-space

  I have assumed that the thickness of the supporting space is small and uniform, because then the intensity of the energy transmitted from a source to any point would vary inversely as the square of the distance, as is the case; whereas if the supporting space was a body of four dimensions, the law would be that of the inverse cube of the distance.

• Hyper-space

  which is usually stated in the form that if a straight line meets two straight lines, so as to make the two interior angles on the same side of it taken together less than two right angles, then these straight lines being continually produced will at length meet upon that side on which are the angles which are less than two right angles.

Equations

• Hyper-space

  \phi(x, y, z) = 0

  The boundary of a solid in ordinary three-dimensional space is the set of points whose co-ordinates satisfy phi(x, y, z) = 0.

• Hyper-space

  \phi(x, y, z, \omega) = 0

  In four dimensions the boundary of a body is the set of points whose co-ordinates satisfy phi(x, y, z, omega) = 0.

Problems

No exercises in this chapter.