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Spherical, Euclidean and Hyperbolic Geometries in Mapping the Brain

Managing Data Using Higher Dimensions

Data is collected for a large sample of individuals where individuals have been assigned to one of two classes by experts. Each individual corresponds to a point in an n-dimensional space where n is the number of measurements recorded for each individual. Mathematics is then used to separate the classes via a plane, similar to the idea of linear regression, but instead of finding a "best fit" line to all of the data, we find the plane that best separates the data into classes.

New individuals are then classified and diagnosed by a computer using the separating plane.

Breast Cancer When a tumor is found, it is important to diagnose whether it is benign or cancerous. In real-life, 9 attributes were obtained via needle aspiration of a tumor such as clump thickness, uniformity of cell size, and uniformity of cell shape. The Wisconsin Breast Cancer Database used the data of 682 patients whose cancer status was known. Since 9 attributes were measured, the data was contained in a space that had 9 physical dimensions. A separating plane was obtained. There has been 100% correctness on computer diagnosis of 131 new (initially unknown) cases, so this method has been very successful.

Heart Disease - Be sure that you have read the text above before performing this activity. View the description of the data. Scroll down to number 7. Use this to identify exactly which attributes were used in the analysis by looking at their abbreviations and then scrolling down to identify the meaning via the complete attribute documentation descriptions.

Find a partner. One of you should read this page as the other follows the directions. View the real-life numerical data that was actually used in the heart disease analysis. Using Select All and then Copy under Edit, copy the numerical data from this link. Open up Word and paste the data into Word. Under Edit, Replace all of the instances of , with ^t . Then under Edit, Select All and then Copy. Open up Excel and paste the data into Excel. It may take a while since there is a lot of data. Each column is a different dimensions worth of data. How many dimensions is this space? Each patient is a different row. How many patients were studied?

This data was the real data that was used to find a separating plane in this higher dimensional data space. New patients have since been diagnosed using this plane. I wanted you to see how you can place it into Excel and to have some experience the actual data that was used. It is not often that one gets the chance to do this, because people rarely make their data sets available to others. You may now quit Excel without saving your file and you should continue reading on the lab web page.

Euclidean Universes

Consider a cube in Euclidean 3-space with the opposite faces glued straight across. This forms a 3-torus. Notice that while we can understand the gluing instructions, we cannot actually glue the space because we would need a 4th physical dimension to do so. As an analogy, think back to 2D Marge who could understand the gluing instructions to form a cube but could not visualize it. She could not even understand how there could be enough space for the gluings to take place. We are in a similar situation when trying to understand the shape of the universe. Yet, we can still understand the properties of this space, understand what it is like to live inside of it, and even devise experiments to test and see whether this is the shape of our universe (similar to Greeks who discovered that the earth was round).

In the above figure, I have drawn a closed straight path which starts from A on the bottom right edge and then hits the middle of the front face at B. It continues from the middle of the back face (since the front face is glued to the back face) and finishes at the middle of the top left edge at a point which is glued to A (via the top and bottom face gluings as well as the left and right side face gluings).

In this picture, we glue pieces of the 3-torus together, by identifying opposite edges. Notice that the gluing of the top and bottom faces and the left and right faces reveal a sphere sitting inside of it. The visualization technique is similar to torus tick-tac-toe where the square above the top right corner was the same as the bottom right square, but instead of squares, we visualize identified blocks. Just above the top right corner, we can draw the figure in the bottom right corner, because they are the same via the gluings. Hence, if we think of a tiling view, then we can see that the 4 parts of the sphere glue together to form a regular sphere.

We can also visualize life inside of a torus universe. This requires the same type of imagination that we used to visualize life in a 2D universe.

The Flatlanders can travel about their flat 2-torus universe without falling off an edge. When looking at a fundamental domain, we must imagine that its edges are glued together in higher dimensions.

Here is a picture of life inside of a 3-torus, with a view that is analogous to the above square that gets glued to form a 2-torus. Here, this cube gets glued to form a 3-torus. Even though the 3-torus is finite, we have the illusion of flying in an infinite space because we never reach an edge. The same thing happens on a 2-torus or on the surface of the sphere because we keep going around and around, passing where we have been before. There are only two stars in this universe but we see each one over and over, like a hall of mirrors.

There are only 10 Euclidean possibilities for the shape of a closed Euclidean universe -- namely, the 3-torus and nine simple variations on it, such as gluing together opposite faces with a quarter-turn or with a reflection, instead of straight across.

In this quarter-turn space, unmarked walls are glued to one another in the simple, straight-across way while the marked side shows that we should glue that side and its opposite side with a rotation by 90 degrees (a quarter of a turn and hence the quarter-turn universe). We identify corresponding squares because squares that are filled in with the same pattern get glued together. The quarter-turn space is a Euclidean universe.

Here is a picture of life inside of a Klein space. We start with a cube and identify 2 of the 3 sets of opposite faces in the usual straight-across way. We glue the 3rd set of faces with a reflection across a line through the center of each face, just as in the Klein bottle. Even though this Euclidean space is finite, we have the illusion of flying in an infinite space because we never reach an edge. There are only two stars in this universe but we see each one over and over. In the Klein space, we fly one way, and see ships in neighboring rows flying in opposite directions. The mirrored images turn, as we do, to fly along paths that seem to cross ours, but they can never hit us - that's impossible in this space.

The above are just a few of the 10 Euclidean possibilities for the shape of a closed Euclidean universe. The others are similarly obtained by gluing together opposite faces of a cube.

Spherical Universes

We glue together opposite sides of this dodecahedron (pentagons get glued to opposite pentagons with a twist (rotation) to make them match up) to obtain a universe that satisfies the laws of spherical geometry. Call Dr. Sarah over to obtain a bag of zometools and use these to build a dodecahedron (1 per group of 3). Then answer the relevant questions on the worksheet.

By gluing together opposite sides of this figure (triangles get glued to opposite triangles and eight sided octagon sides get glued to the opposite octagon side) we obtain another spherical universe.

The number of spherical possibilities are infinite, but have been classified completely. Dr. Sarah's research relates to spherical universes.

Hyperbolic Universes

By gluing together corresponding sides of this 18 sided figure (for example, the pentagon faces get glued together), we obtain a hyperbolic universe. This example was discovered by Jeff Weeks.

There are infinitely many possible topologies for a finite hyperbolic three-dimensional universe. Their rich structure is still the subject of intense research and the classification is still an open problem today.

According to relativity, space is a higher dimensional dynamic medium that can curve in one of three ways, depending on the distribution of matter and energy within it. Because we are embedded in space, we cannot see the flexure directly but rather perceive it as gravitational attraction and geometric distortion of images. To determine which of the three geometries our universe has, astronomers have been measuring the density of matter and energy in the cosmos.

Many cosmologists expect the universe to be finite, curving back around on itself. Mach inferred that the amount of inertia a body experiences is proportional to the total amount of matter in the universe. An infinite universe would cause infinite inertia. Nothing could ever move. In addition to Mach's argument, there is preliminary work in quantum cosmology, which attempts to describe how the universe emerged spontaneously from the void. Some such theories predict that a low-volume universe is more probable than a high-volume one. An infinite universe would have zero probability of coming into existence [see "Quantum Cosmology and the Creation of the Universe," by Jonathan J. Halliwell; Scientific American, December 1991]. Loosely speaking, its energy would be infinite, and no quantum fluctuation could muster such a sum.

Historically, the idea of a finite universe ran into its own obstacle: the apparent need for an edge. Aristotle argued that the universe is finite on the grounds that a boundary was necessary to fix an absolute reference frame, which was important to his worldview. But his critics wondered what happened at the edge. Every edge has another side. So why not redefine the "universe" to include that other side? German mathematician Georg F. B. Riemann solved the riddle in the mid-19th century. As a model for the cosmos, he proposed the hypersphere--the three-dimensional surface of a four-dimensional ball, just as an ordinary sphere is the two-dimensional surface of a three-dimensional ball. It was the first example of a space that is finite yet has no problematic boundary.

One might still ask what is outside the universe. But this question supposes that the ultimate physical reality must be a Euclidean space of some dimension. That is, it presumes that if space is a hypersphere, then that hypersphere must sit in a four-dimensional Euclidean space, allowing us to view it from the outside. Nature, however, need not cling to this notion. It would be perfectly acceptable for the universe to be a hypersphere and not be embedded in any higher-dimensional space. Such an object may be difficult to visualize, because we are used to viewing shapes from the outside. But there need not be an "outside."

Experiments

Similarly, we can use indirect ways of trying to gain an appreciation for the counterintuitive behavior of 4D objects.

A hypercube is one of the "easiest" 4D objects to try and understand. Yet, it cannot be the shape of our universe since it has an edge, which means that there would have to be something on the other side of the edge.

References -- Adapted from excerpts taken from: