The Geometry of a Sperical Cow
August 27. Welcome home. Grab a coffee and let’s get smarter together.
Just mentioning geometry might cause you to break out in cold sweat. Nostalgia has not yet romanticized rulers and compasses for you. Be brave my friends. This will be more of a journey into the history of the subject for the sake of our dear spherical cow.
The foundations for geometry were laid – probably by women – far, far in the past because you simply cannot weave, sew, or potter without some understanding of e.g. the angles in a square or the length of the circumference of a circle. It was on Euclide to write down what was known and what we know today as Euclidean geometry. The kind of tabletop geometry that can all be reduced to 2D thinking – you can unfold any cube or whatever so its sides form a pattern in 2D – and that we encounter in 3D model of the world our mind produces for us.
Just as an aside because I can’t stop myself being an insufferable know-it-all: As Poverty Point, LA demonstrates, the indigenous population of the Americas had their own version of geometry that they shared across the continents. What begs the question how many other ways to thing about space have there been that European conquer has eradicated.
Anyways, back to Euclidean geometry. One of the rules Euclide stipulated in his work was that the distance between parallel lines remains the same into infinity. Imagine a line with two lines sticking out perpendicularly. According to Euclide they’ll never meet.
Well, now imagine the same two lines sticking out perpendicularly from the equator of a sphere. They will of course meet at the poles of the sphere. Everyone knew it. Just as everyone knew that the angles of a triangle drawn on the surface of a sphere add up to more than 180° or that the circumference of a circle drawn on a sphere was smaller than 2πr. These are all things that shouldn’t happen according to Euclide. But for something like two thousand years we just rolled with it. No one considered creating a whole new branch of geometry just for a sphere.
Along came Lobachevski and Bolyai and thought to themselves, if its possible that parallel lines curve to meet like on a sphere, who says they can’t diverge? This happens if your two dimensional plane itself is curved to resemble almost a saddle or Pringles potato chip, but not quite. Because, and that will be important, an exact hyperbolic plane exists only in our minds. Other than a sphere we can’t recreate such a space as a two-dimensional surface embedded in the space where we live.
That’s important because it requires a huge shift in the way we think about space. The way our mind suggests to us is that we put ourselves apart from the ‘object’ and observe its extrinsic properties that it inherits from being embedded in a larger space. Hyperbolic planes don’t exist in any larger space, so we can’t do that. We must study its intrinsic geometrical properties by understanding ourselves as a part of it.
Here an example that might make the difference clear. When looking at an empty toilet paper role, we recognize it as a cylinder. That’s the extrinsic impression of an observer who lives in a 3D space thanks to their mind. Now, imagine yourself being a part of the cardboard and walking and walking and walking. The outside observer would watch you walk circles around the cylinder. But you, being part of the cardboard, perceive it as a flat surface. That’s the intrinsic view.
The quantum space we live in resembles hyperbolic space in the aspect that we can’t observe it. All we can do, is study it while we accept being a part of it.
Two things to take from this story:
1. Humans knew for approx. two thousand years that Euclidean geometry wasn’t the whole story. Instead of exploring what was missing we rather tried to make it work somehow anyway. This reminds me of all the proof we got that neither Hobbs nor Rousseau were right with their assumptions about the human nature, nevertheless we put all our energy and literally sweat, blood, tears, and lives into proving esp. Hobbs right.
2. The model of the world that our mind produces, mostly by using creativity to fill our the blanks in the fishnet, doesn’t only suggest to us that we are an observer when it comes to shapes in space. Since we identify with the self that the mind cuts out of the tapestry of the generated model, it’s our constant POV. This distance between us and the physical universe does not exist. The physical universe is one large wave function – vibrations of energy – and we are a tiny part of it. We need to start with an intrinsic view to learn the truth about us – the spherical cow that leaves the complication that is the mind with its eccentric viewpoint out of it – and explore from there. Maybe we even rediscover our sixth sense, the sense for energies, on the path.
Turn your curiosity back on. Explore. Make a habit of exploring. Let’s turn exploring into the norm again, for all.
Like, comment, share – and follow, lest you miss a single opportunity to improve yourself and society. Once you do, we’ll meet again. At the coffeepot.
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