Geometries of the Universe: The Math of Knowledge Advancement
What can math be used for? Here's a wise answer: two basic forms of geometry are used in almost every engineering project and every physics discovery that has ever been made.
Back in 8th grade, I hated math. Everyone hated math. Maybe the kid who kept their calculator (or slide rule, for you vintage readers) in a case didn't hate math. That kid probably became an engineer. Or a physicist. (Confession: I was later the kid with the fancy calculator.) Our rallying cry as math-haters was, "When are we ever going to use this?!" Here's a wise answer: two basic forms of geometry, learned before high school, are used in almost every engineering project and every physics discovery that has ever been made.
Greek mathematicians, notably Euclid and Pythagoras (of middle school algebra infamy), laid out the first geometry of the world. They thought of things in terms of shapes made of lines and curves. Their most important discovery was a way to tell how far apart things are:
Take any two places (A and B) and draw a line through each place such that the lines meet (C) at a 90 degree angle. The distance from A to B, squared, is always equal to the distance B-C squared plus the distance from C back to A, squared. (This is the infamous Pythagorean theorem.) This language is perfectly accurate for flat, still surfaces. Notice however, that it only deals in distances between things, not their absolute position. Euclid says "B is five miles north of A" not "B is at 2 Water Lane, Woolsthorpe".
Descartes wanted a way to make the points A, B and C refer to absolute things so that anyone anywhere can perform the same measurements. Latin, Chinese, Hebrew and English are all languages of words to catalog or refer to concepts. They are phone books that assign words to ideas. Similarly, the math of Descartes is a phone book, but to assign numbers to places in space. This is called Cartesian geometry. In this language, the Pythagorean Theorem is written like this:
Where A, B and C are all coordinate numbers, like (0,0) or (-3,5) that you stick into the formula. Euclid would have made you draw lines and geometric shapes and connect them all with theorems!
Descartes's world is an enormous ream of numbered graph paper. You start with zero somewhere, and then you follow perpendicular lines in all directions. Euclid's relative distances are replaced by numbers that tell you where you start and where you end and where you are everywhere in between, relative to the entire world. This mathematical machinery is valid for most experiences in day to day life.
Centuries later, Einstein came along and changed everything. His conclusion that the speed of light is constant, and his fitting of experiments to theory demanded a new geometry. In this geometry, objects always move at the speed of light through four-dimensional space-time. The math was invented by Hendrik Lorentz, a brilliant mathematician and physicist of the late 19th century. Lorentzian geometry is much harder to explain, but you can think of the graph paper of Descartes as actually distorted, or squished, like a cardboard carton being smashed:
Cartesian geometry is the black perpendicular lines; Lorentz geometry is the green and red lines. (Source: IEP)
Second, distance rules change a bit, so you have to modify the Pythagorean theorem:
Where x,y,z,t are the distances you've moved in space (x,y,z) and time (t) and c is the speed of light.
Curved spacetime geometry:
After Einstein revolutionized the geometry by which we measure the universe in 1905, he did it again in 1916, when he completed the theory of general relativity. General relativity is so difficult and so complex, that we only know of a few correct answers to its equations. Luckily, however, Einstein realized that the curved universe looks flat if you look at a small enough area of it. This is just like how the earth appears flat to us, standing upon it. Physicists thus work in Lorentz geometry most of the time, and then use the difficult curved geometry to translate between one almost flat "Lorentzian" place and another.
Four increasingly sophisticated geometrical descriptions of the universe, as created by three millennia of human minds. How much deeper will the rabbit hole go?
Note: phone book analogy borrowed from "The Phone Book": Misner Thorne & Wheeler's Gravitation, a book on general relativity with the authority and heft of a phone book but far greater beauty.
The way that you think about stress can actually transform the effect that it has on you – and others.
- Stress is contagious, and the higher up in an organization you are the more your stress will be noticed and felt by others.
- Kelly McGonigal teaches "Reset your mindset to reduce stress" for Big Think Edge.
- Subscribe to Big Think Edge before we launch on March 30 to get 20% off monthly and annual memberships.
Three scientists publish paper proving that not Venus but Mercury is the closest planet to Earth
- Earth is the third planet from the Sun, so our closest neighbour must be planet two of four, right?
- Wrong! Neither Venus nor Mars is the right answer.
- Three scientists ran the numbers. In this YouTube video, one of them explains why our nearest neighbour is... Mercury!
The blood of horseshoe crabs is harvested on a massive scale in order to retrieve a cell critical to medical research. However, recent innovations might make this practice obsolete.
- Horseshoe crabs' blue blood is so valuable that a quart of it can be sold for $15,000.
- This is because it contains a molecule that is crucial to the medical research community.
- Today, however, new innovations have resulted in a synthetic substitute that may end the practice of farming horseshoe crabs for their blood.
The distance between the American dream and reality is expressed best through literature.
- Literature expands our ability to feel empathy and inspires compassion.
- These ten novels tackle some facet of the American experience.
- The list includes a fictional retelling of the first Native American to graduate from Harvard and hiding out in inner city Newark.
SMARTER FASTER trademarks owned by The Big Think, Inc. All rights reserved.