Manual An atlas of the smaller maps in orientable and nonorientable surfaces

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Since each change was so mild, you must never have changed the fundamental shape of the object.

Discrete Mathematics and Its Applications

The branch of mathematics concerned with the study of shape and space is called topology from ancient Greek topos, meaning place. An oft-repeated and never funny joke is that a topologist cannot tell a coffee mug from a donut. This is not true. But it is true that mathematicians abstract the sensory qualities from a thing the heft of the mug, the sweetness of the donut.

And if both mug and donut were made of a perfectly malleable substance, you might expand the base of the mug to fill up its volume, then shrink the resulting thick cylinder down until it's the same width as the handle, leaving a torus. Two surfaces A and B have the same shape if you can draw a faithful map of A on B.

In other words, there is a continuous correspondence between the points of A and the points of B--an assignment to every point of A exactly one point of B in such a way that any smooth path you draw between two points on A maps to a smooth path on B, and vice versa. In this sense, the Mercator projection is not a map of the earth.

To begin with, two points, the North and South poles, are missing entirely. Moreover, on the real earth one can sail from San Francisco to Japan through the Pacific. On the Mercator projection, the ship hits the edge of the map. If you start with a object and deform it, squeezing, pinching, pulling, etc, then whatever you end up with comes with a map already made: the surface of the original lies distorted on the final product.

While the earth unbeknownst to our emperor is not a perfect ball, its surface not a perfect sphere, they are, topologically, the same shape: push in the mountains, pull out the valleys, smooth it out. We can now compare physical surfaces to infinitely thin abstract ones. We can jump scale from maps and atlases to planets—just write down a correspondence of points.

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Consider the emperor's cube stitched together out of the atlas's pages. Place it at the center of a Japanese lantern, a thin white paper sphere, with a very bright light inside the cube. The projection is our one-to-one correspondence, the image of the cube exactly covers the sphere; the cube is the same shape as the sphere. How could we prove that no possible correspondence can exist between two surfaces?

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If one surface fits in the palm and we live inside the other, if one is presented as pages in an atlas and the other as a habitat, we might have to be rather clever to find the right correspondence. We need an invariant, something which can be computed about a surface that doesn't depend on how it's presented to us, something that can prohibit the existence of any continuous correspondence and tell us: these really are different. A first difference: the disk is not the sphere. The disk has a boundary, an edge, and the points on the edge are special. To see this, imagine yourself living in the surface.

Imagine that the thin surface is sandwiched between two sheets of glass, and you are a paramecium, a single-celled organism swimming sideways. When you come to an edge, you have to stop--points on the boundary of the disk have the special property that some directions of travel are forbidden. At every point of the sphere, however, you have a full range of motion. There can be no continuous correspondence between the disk and the sphere because the point on the sphere ending up on the boundary of the disk would have to be ripped away from its neighbor.

Tearing is very discontinuous. This gives us our first invariant, the number of components in the boundary of a surface or the number of holes it has cut into it. A sphere with two holes at opposite poles is the same shape as a sphere with two holes next to each other is the same as a hollow tube. We can also distinguish finite surfaces from infinite surfaces like the vast plane that Euclidean geometry takes place on. We're assuming that all the surveyors returned with good news, so the theoretical earth we're trying to map is finite.

It's orientable if it has two sides like the sphere , which we usually call the in-side and the out-side. We can also define orientability intrinsically, for people who live in the surface, not on it. People like our paramecium. Now TK can wander around through the surface, taking some convoluted path, then return to where he started. For many paths, he still looks like JW upon his return, cilia twisting clockwise.

Perfect Shapes in Higher Dimensions - Numberphile

If there is a path TK can take and come back reversed from his twin, then the surface is nonorientable. Otherwise, it's orientable. This implies the extrinsic definition by the right-hand rule.

Orientability - Wikipedia

The surfaces of things, of thick 3-dimensional bodies, are necessarily orientable. The body itself defines the inside, which cannot be reached traveling on the outside.

Surfaces of things are also finite, and have no boundary--if you try to cut a hole in the surface, you just make a depression extending the surface inwards. So to classify surfaces of things, like possible shapes for the earth, we need only consider closed, finite, orientable shapes, like the sphere and the torus. To distinguish the sphere and the torus, we need a more sophisticated invariant called the Euler characteristic. To define it, we'll take a brief digression into graph theory.

A graph is a bunch of vertices connected by a bunch of edges. We will ask that our graph be connected, so it can't be split into two disconnected pieces. We will also ask that the graph be planar: edges don't cross. They only meet at vertices. Disconnected graphs are forbidden. Edges should not cross. On a planar graph, we can also count faces.

Michael La Croix

Those are the regions of the plane bounded by edges, the areas that would get colored in MS paint if you put the little paint bucket over them and clicked. The outside counts as one big face, since when you click, it's colored. This graph has 6 vertices, 8 edges, and 4 faces. Its Euler characteristic is 2.

Theorem: The Euler characteristic of a connected planar graph is always 2. Proof: Start drawing the graph. Ask Question. Asked 4 years, 1 month ago. Active 4 years, 1 month ago. Viewed times. PyRulez PyRulez 5, 2 2 gold badges 24 24 silver badges 75 75 bronze badges. In particular, it is invariant under homeomorphisms. Rhys Rhys 3, 10 10 silver badges 23 23 bronze badges. For a general topological space, it has no meaning.

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