In the long ago times, mathematics was a leisure activity. The ancient Greeks dabbled in the subject to pass the time. This was like some people now enjoy jigsaw puzzles.

Then came **MATH TERROR**!

In Antarctica, Archaean mythical resemblances were found to correspond to the evilly famed plateau of Leng in the primal writings. Monstrous perversions of geometrical laws were seen. “There were truncated cones, sometimes terraced or fluted, surmounted by tall cylindrical shafts here and there bulbously enlarged and often capped with tiers of thinnish scalloped discs; and strange, beetling, table-like constructions suggesting piles of multitudinous rectangular slabs or circular plates or five-pointed stars with each one overlapping the one beneath,” wrote H.P. Lovecraft in, *At The Mountains of Madness*.

In *The Cursed Scrolls of R’lyeh*, by Arthur Wilcox Hodgkins, we find, “… the geometry was wrong— abnormal, non-Euclidean, and loathsomely redolent of spheres and dimensions not of our own!” [1]

It was, in other words, the Non-Euclidean Geometries!

As nearly as I can discern from poring over the musty texts, there are two types of these Non-Euclidean Geometries: (1) Spherical geometry, the geometry of the two-dimensional surface of a sphere; and (2) hyperbolic geometry, also called Lobachevskian geometry and named after the Mad Russian, Nikolai Ivanovich Lobachevsky (image top).

The Lobachevskian hyperbolic geometry involves something called the sectional curvature. The sectional curvature comes to us by way of Carl Friedrich Gauss’s theory of curved surfaces in space. [2] The realm of curvature began with the ancient Greeks who noticed: (1) lines do not curve; and (2) every point on a circle curves the same amount. Aristotle expanded on this and declared three kinds of loci: straight, circular, and mixed. [3]

One Nicole Oresme in the 14th Century discovered what he called “curvitas”, a specific measure of twist. [3]

Rene Descartes in his coordinate system linked algebraic equations with curvatures such as the circle, parabola and hyperbola. [3] This latter – hyperbola – seems to connect with the term “hyperbolic” in the dreaded Lobachevskian hyperbolic geometry.

“Because Euclidean geometry and hyperbolic geometry are both consistent and are in an environment with a small sectional curvature very similar an observer will have a hard time determining whether his environment is Euclidean or hyperbolic. We also cannot decide if our world is Euclidean or hyperbolic.” [4]

Only the brave will have read up to this point in today’s Ersjdamoo’s Blog entry. The rest will have run away, rationalizing their fear with notions such as “I don’t have time for this” (even though the Ersjdamoo’s Blog entries are typically short, under about 5K).

Whence comes this Math Terror? Remember that math used to be an enjoyable leisure activity, like doing jigsaw puzzles. The math terror comes from the assembly line factory technique of education. No longer is math a leisure. Instead the assembly line “class” is confined in a room and tyrannized by a 50-minute clock. “You have 50 minutes. Begin now. The clock is ticking. Tick-tick-tick. Solve the problems – or else!”

Suppose the school offered “Jigsaw Puzzles 101.” The Forrnier-Dufus theorem states that the four corners of the jigsaw puzzle must first be found. But the Gaussian-Lobachevsky expands this into circular jigsaw puzzles and the Forrnier-Dufus is rendered superfluous. The clock is ticking. Tick-tick-tick. You have 50 minutes to solve the jigsaw puzzle. Begin now.

What would you learn by this method of instruction for the “Jigsaw Puzzles 101” class? Answer: You would learn to hate and fear jigsaw puzzles!

Mathematics is a universal language. For purposes of peace, it has been the hope that some sort of language like Esperanto, a constructed international auxiliary language, an easy-to-learn, politically neutral language, would transcend nationality and foster peace and international understanding between people with different languages. [5] But we already have the universal language, without the Esperanto! The universal language which can help promote peace is mathematics, and we are teaching people to hate and fear it.

——- Sources ——-

[1] “The Cursed Scrolls of R’lyeh”, by Arthur Wilcox Hodgkins. Part of The Xothic Legend Cycle of Lin Carter — follow-up to “The Horror in the Gallery.” http://tinyurl.com/lejteun

[2] “What Is Curvature?” http://www.math.washington.edu/~lee/Books/Riemannian/c1.pdf

[3] “Curvature of Surfaces in 3-Space”, by Michael Garman and Jessica Bonnie.

[4] “Hyperbolic geometry”, Wikipedia, March 18, 2015.

[5] “Esperanto”, Wikipedia, March 18, 2015.