Famous Geometry Quotes

The long triangular grooves on the car had been formed within the death of an unknown creature, its vanished identity abstracted in terms of the geometry of this vehicle. How much more mysterious would be our own deaths, and those of the famous and powerful?

Key traits are a large x-height, strict geometry, and the infamous leaning letters and ligatures that allow extra tight fitting words. These glyphs are often abused, leading Ed Benguiat to famously declare, “The only place Avant Garde looks good is in the words ‘Avant Garde.

No one is alone in this world. No act is without consequences for others. It is a tenet of chaos theory that, in dynamical systems, the outcome of any process is sensitive to its starting point-or, in the famous cliche, the flap of a butterfly's wings in the Amazon can cause a tornado in Texas. I do not assert markets are chaotic, though my fractal geometry is one of the primary mathematical tools of "chaology." But clearly, the global economy is an unfathomably complicated machine. To all the complexity of the physical world of weather, crops, ores, and factories, you add the psychological complexity of men acting on their fleeting expectations of what may or may not happen-sheer phantasms. Companies and stock prices, trade flows and currency rates, crop yields and commodity futures-all are inter-related to one degree or another, in ways we have barely begun to understand. In such a world, it is common sense that events in the distant past continue to echo in the present.

This special was followed one month later by “Bart the Genius.” This was the first genuine episode of The Simpsons , inasmuch as it premiered the famous trademark opening sequence and included the debut of Bart’s notorious catchphrase “Eat my shorts.” Most noteworthy of all, “Bart the Genius” contains a serious dose of mathematics. In many ways, this episode set the tone for what was to follow over the next two decades, namely a relentless series of numerical references and nods to geometry that would earn The Simpsons a special place in the hearts of mathematicians.

I was somewhat surprised, therefore, to find that my college teachers—famous academics and composers—inhabited an entirely different musical universe. They knew nothing about, and cared little for, the music I had grown up with. Instead, their world revolved around the dissonant, cerebral music of Arnold Schoenberg and his followers. As I quickly learned, in this environment not everything was possible: tonality was considered passé and “unserious”; electric guitars and saxophones were not to be mixed with violins and pianos; and success was judged by criteria I could not immediately fathom. Music, it seemed, was not so much to be composed as constructed—assembled painstakingly, note by note, according to complicated artificial systems. Questions like “does this chord sound good?” or “does this compositional system produce likeable music?” were frowned upon as naive or

Descartes’s worldview makes us spiders at the center of an enormous web not of our making. Or in his other famous formulation, we are the ghosts in the machine: souls in a world machine that operates inexorably and impersonally according to the laws of geometry and mechanics, while we operate the levers and spin the dials.

use four-dimensional Riemannian geometry for his famed theory of relativity. Within seven decades, Theodr Kaluza at the University of Königsberg, Germany, would use five-dimensional Riemannian geometry to integrate both gravity and light. Light is now viewed as a vibration in the fifth dimension. Oskar Klein made several improvements, including the calculation of the size of the fifth dimension—the Planck length, which is 10-33 centimeters, much too small to detect experimentally. One hundred thirty years after Riemann’s famous lecture, physicists would extend the Kaluza-Klein constructs to develop ten-

There is geometry in the humming of the strings; there is music in the spacing of the spheres.

Inventor Buckminster Fuller attributed his invention of the geodesic dome to his early rejection of both the standard x, y, z and polar systems in favor of a tetrahedral paradigm. Einstein similarly rejected Euclidean geometry for a non-Euclidian formulation that gives rise to his famous description of space and time curving in relativistic gravitational fields. Both Einstein and Fuller understood explicitly that Euclidean geometry is only one version of the world. Non-Euclidian geometries, spherical geometries, and many other mathematical formulations of space exist, each providing a different set of patterns for the use of inventors, builders, artists, and other innovators. The problem is that we can’t use what we don’t know. Our pattern-recognizing ability benefits from practice with these different versions of space, just as it benefits from familiarity with different forms of hopscotch.

Passyunk Avenue (pronounced pashunk by the locals) cuts a rude swath across an otherwise orderly grid of streets in South Philadelphia. Except for Passyunk (and Moyamensing) Avenue, the neighborhood is composed of a uniform matrix of numbered and named streets—one big street followed by two little streets. Viewed on a map, they form ninety-degree angles and predictable intersections. Passyunk Avenue, or simply Passyunk, is the great disruptor of this comforting geometry. Irregular and meandering, its slashing path intersects with the more obedient byways. Together they form a unique gridwork of inconvenient crossings and odd angles. The cumulative result is one of strangely shaped buildings. Their pointy corners puncture curious cells of dead space—the spaces between. While born of necessity, the resulting architecture created by these acute angles also manages to be strangely beautiful, an exotic visage in a sea of pretty faces. If you’ve ever seen the famous photo of Sophia Loren giving the side-eye to Jayne Mansfield, that’s Passyunk—South Philly’s middle finger to white bread Center City.

Thus Plato offers one version of a philosophical poverty, by which wisdom alienates one from conventional ideals and makes one indifferent to worldly concerns. The Cynics philosophized in the same general rubric, though obviously details differ significantly. One notable difference between the two is the value placed upon learning and science. Unlike his Platonic counterpart, the Cynic rejects arithmetic, geometry, dialectic, and the rest as superfluous distractions from the "one big" requirement of self-knowledge. So in the famous anecdote, when Plato and his followers have defined man as the "featherless biped," Diogenes rushes into the Academy with a plucked chicken, crying, "Here is Plato's human being!" For the Cynic, logical exercises, definitionmaking, and the like are not preparatory to the vision of some Good or intuition of eternity. All such talk is a form of pride, a strategy to overawe others, and contributes less to the good life than does a healthy skepticism. Yet, like Plato, the Cynics also travel along the Eleatic Way of Truth and shun what they ridicule as the Way of Seeming. For what can be said truly? In short, the Cynics are profoundly skeptical about the possibility of almost all knowledge.

Flower of life: A figure composed of evenly-spaced, overlapping circles creating a flower-like pattern. Images of the Platonic solids and other sacred geometrical figures can be discerned within its pattern. FIGURE 3.14 FLOWER OF LIFE The Platonic solids: Five three-dimensional solid shapes, each containing all congruent angles and sides. If circumscribed with a sphere, all vertices would touch the edge of that sphere. Linked by Plato to the four primary elements and heaven. FIGURE 3.15 PENTACHORON The applications of these shapes to music are important to sound healing theory. The ancients have always professed a belief in the “music of the spheres,” a vibrational ordering to the universe. Pythagorus is famous for interconnecting geometry and math to music. He determined that stopping a string halfway along its length created an octave; a ratio of three to two resulted in a fifth; and a ratio of four to three produced a fourth. These ratios were seen as forming harmonics that could restore a disharmonic body—or heal. Hans Jenny furthered this work through the study of cymatics, discussed later in this chapter, and the contemporary sound healer and author Jonathan Goldman considers the proportions of the body to relate to the golden mean, with ratios in relation to the major sixth (3:5) and the minor sixth (5:8).100 Geometry also seems to serve as an “interdimensional glue,” according to a relatively new theory called causal dynamical triangulation (CDT), which portrays the walls of time—and of the different dimensions—as triangulated. According to CDT, time-space is divided into tiny triangulated pieces, with the building block being a pentachoron. A pentachoron is made of five tetrahedral cells and a triangle combined with a tetrahedron. Each simple, triangulated piece is geometrically flat, but they are “glued together” to create curved time-spaces. This theory allows the transfer of energy from one dimension to another, but unlike many other time-space theories, this one makes certain that a cause precedes an event and also showcases the geometric nature of reality.101 The creation of geometry figures at macro- and microlevels can perhaps be explained by the notion called spin, first introduced in Chapter 1. Everything spins, the term spin describing the rotation of an object or particle around its own axis. Orbital spin references the spinning of an object around another object, such as the moon around the earth. Both types of spin are measured by angular momentum, a combination of mass, the distance from the center of travel, and speed. Spinning particles create forms where they “touch” in space.