Cissoid of Diocles

The Cissoid of Diocles is an effort by ancient greek mathematician Diocles to solve the problem of doubling the cube.  Mythologically, this doubling was a challenge given by the gods to the Athenians to make an alter double the size of an original.  The Athenians constructed an alter with length, width, and height twice the original – but the gods were unimpressed because, by size, they meant volume, not the dimensions.  The volume of the resulting alter was actually eight times the original, and not the prescribed doubling.  The gods are picky.

Anyway, Diocles’ effort to solve this problem, loosely with a nod to the Euclidian restraints of simply a straight-edge and compass, is a graph with a “ivy-shaped” set of curves (hence the name) that establish a relationship of two mean proportionals to a ratio.

Diocles was fascinated by conic sections, and his notes, “On Burning Mirrors”, he explores the focal points of parabolas in search of an optimal focus.

Cissoid of Diocles

spiral

Archimedes’ Spiral

Archimedes of Syracuse (c. 287 B.C. – c. 212 B.C.) published an examination of the phenomenon of spirals in 225 B.C. called “On Spirals“.  In this, he stated his spiral defining equation r=a+bø whereas “a” is the parameter and “b” controls the expansion of the subsequent rotations.   In his own words, “If a straight line one extremity of which remains fixed is made to revolve at a uniform rate in a plane until it returns to the position from which it started, and if, at the same time as the straight line is revolving, a point moves at a uniform rate along the straight line, starting from the fixed extremity, the point will describe a spiral in the plane.” Heath, Thomas Little (1921), A History of Greek Mathematics, Boston: Adamant Media Corporation, p. 64,ISBN 0-543-96877-4, retrieved 2008-08-20

What I Learned About Mathematics Today: Characteristics of Positive Integers

Any two positive positive integers sum to a positive integer and multiply to a positive integer.  This is being “closed” to addition and multiplication.

Though there is a finite number of integers that sum to, or are multiples of, a positive integer, the number that subtract to, or are divisors of, is infinite(!)

[courtesy of “A Year of Mathematics” calendar by Steven J. Alexander]