Titlebook: Concepts & Images; Visual Mathematics Arthur L. Loeb Book 1993 Springer Science+Business Media New York 1993 design.mathematics.synergetics

要塞

Points and Regions,In this chapter we summarize the various arrays of discrete points and the various tessellating polygons which we have encountered in the preceding chapters, and introduce some others. Notably, there is a one-to-one correspondence between some of the discrete points and the polygons.

无效

A Look at Infinity,Four bugs are located at the four corners of a square. Each looks at a bug nearest to it in a clockwise direction. Each moves toward that neighbor, all four bugs moving at the same speed at any given moment, although that speed does not necessarily remain constant in time.

FECT

An Irrational Number,The bugs studied in the previous chapter generated a curve which makes a constant angle, namely 45° with the direction toward the origin (the radial direction). We could have used six bugs at the corners of a regular hexagon, in which case they would have travelled at 60° to the radial direction.

Debate

The Notation of Calculus,In Chapters XIII and XIV we dealt with issues of discrete and continuous structures, rational and irrational numbers, and recognized the relationships between them. These are actually the fundamental concerns of calculus; if they are understood, then the remainder of calculus is essentially a question of notation.

善辩

Tessellations and Symmetry,t overlap or spaces in between is said to be ., a term derived from the Greek word ., or tessera, a tile. Principally, we shall concern ourselves here with the problem of covering a plane with mutually identical tiles; in Chapter XX we shall deal with a particular . of tiles.

王得到

A Diophantine Equation and its Solutions, Diophantes of Alexandria, who is presumed to have discovered them. In general, all variables in such an equation are to be rational; in our case they are integers. Although in general one cannot solve a single equation in three variables, the restriction that the variables be integers limits us to a finite number of solutions.