an infinity between.

If viewed as is this diagram may appear to contradict an earlier post explaining broad-brush set theory and the notions of cardinality and infinity -- all of which occurred to me immediately after having posted this. Its cool colors and overall treatment of completeness appeal to me. 

In terms of "size" ℚ = ℤ despite what the diagram may suggest. That is to say, the integers form a one-to-one correspondence with the rationals. Cantor proved that for any member of ℤ a unique ℚ may be matched to it, so the two sets are of equal "size." The same method attempted on the irrationals shows that it is impossible (epexegesis: earlier post) to even list them. Therefore, the infinitude of the reals is larger than the infinity of both ℤ and ℚ.

intercomparing circular and hyperbolic trigonometric functions.

The circular trigonometric identity cos²(u) + sin²(u) = 1 describes each point on the perimeter of the circle x² + y² = 1 as a function of angle u. 

In a sense the hyperbolic trigonometric identity cosh²(u) - sinh²(u) = 1 mimics this approach by substituting x² - y² = 1 as a function of hyperbolic sector u.

square roots as points of intersection of two circles.

Credit to whoever cast this mathemagical spell at Perspective Infinity. 

A circle of radius r₃ lies on the plane.


Within the circle are inscribed six others: three white ones of radius 1/r₃, which is the curvature of the white circle; two orange ones of radius (r₃² + 2)/2r₃; and a red one of radius . The point at which the red intersects the orange is the square root of r₃.

recanting ideas of infinity

A short video on infinity and Cantor's continuum hypothesis.

the law of cosines by Sydney Kung.

Summary here: http://www.pleacher.com/mp/mlessons/trig/lawcos.html.