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abstraction > mathematical concept > geometrical object

geometrical object
subject			fact

geometrical object	is a kind of mathematical concept
	has dimensions

Kinds of geometrical object :

• conic section (12 kinds, 36 facts)
• fractal (4 facts) - A geometric figure in which a pattern is repeated ad infinitum on smaller and smaller scales. A classic example is Von Koch's snowflake, for which the construction begins with an equilateral triangle. Trisect each side, and replace the middle section by two sides of a smaller equilateral triangle, bulging outward. The snowflake is obtained by repeating this process for each side of the resulting figure, then for each side of the subsequent figure, and continuing forever., A pattern that repeats itself or nearly repeats itself on many different scales of magnification. For example, suppose that some ink on a piece of paper appears to form a star. If you look at the piece of paper with a magnifying glass, you see that the dark areas are not solid black, but are formed of tiny stars themselves. If you look at one of these small stars with a microscope, you see that the dark areas of each of the tiny stars is formed from an arrangement of even tinier stars. Such a repeating pattern of stars would be called a fractal.
• line (3 kinds, 9 facts) - A geometrical object with one dimension
• plane (10 kinds, 25 facts)
• point (40 kinds, 165 facts) - A geometrical object with zero dimensions, a location and possibly a time
• sphere (1 kind, 6 facts) - The outer surface of a ball. The surface of a familiar three-dimensional ball has two dimensions (which can be labeled by two numbers such as "latitude" and "longitude," as on the surface of the earth). The concept of a sphere, though, applies more generally to balls and hence their surfaces, in any number of dimensions. A one-dimensional sphere is a fancy name for a circle; a zero-dimensional sphere is two points (as explained in the text). A three-dimensional sphere is harder to picture; it is the surface of a four-dimensional ball.
• torus (4 facts) - The topological name for the shape of a donut. While a donut is a two-dimensional surface in a three-dimensional space, the torus can be generalized to higher numbers of dimensions., The two-dimensional surface of a doughnut.