### Adapted from excerpts taken from:

• Davide Cervone's materials
• Cathy Gorini Geometry at Work
• David Henderson Experiencing Geometry in the Euclidean, Spherical, and Hyperbolic Spaces
• Jean-Pierre Luminet, Glenn D. Starkman and Jeffrey R. Weeks Is Space Finite?, Scientific American
• Diane Martindale Road Map for the Mind: OLD MATHEMATICAL THEOREMS UNFOLD THE HUMAN BRAIN, Scientific American
• Jeff Weeks Exploring the Shape of Space

### Review and Introduction

During last week's lab, we saw that a 2-D universe can be Euclidean, spherical or hyperbolic.
• The surface of a sphere satisfies the laws of spherical geometry from project 2.
• The torus (donut) and Klein bottle satisfy the laws of Euclidean geometry from high school.
• The 2-holed torus satisfies the laws of hyperbolic geometry like Escher's angels and devils.
• We'll explore theories about the geometry and shape of our universe. We will also continue to see connections to art, philosophy, physics, astronomy, geography, and visualization. Because our brains are wired to see 3-D (by using layering 2-D slices ), if you are properly engaging the material, these ideas should stretch the limits of your imagination.

In order to save paper, I have not printed this lab. I recommend that you work in a group using at least two computers, with one person reading these directions from their computer as the other performs the activities.

• We'll begin lab by discussing an update on Robert Kirshner's Supernovae results related to whether brightness=1/distance2

Answer the following questions on a sheet of paper.

• Question 1 Describe Rob Kirshner's Supernovae distance/brightness experiments.
• Question 2 Does the data give us a convincing "answer" about the geometry of the universe?
• Question 3 Give one critique of the experimental design.

### Real-life Applications of Related Material

• We have previously discussed the applications of spherical geometry to architecture, maps, Einstein's theory of relativity, and travel on the earth. Did you know that Mercury's orbit about the Sun is slightly more accurately predicted when Hyperbolic Geometry is used in place of Euclidean Geometry? In addition, artists and mathematicians have been creating hyperbolic coral reefs:

Hyperbolic reef

### Spherical, Euclidean and Hyperbolic Geometries in Mapping the Brain and the Internet

All those folds and fissures make life difficult for a neuroscientist: they bury two thirds of the brain's surface, or cortex, where most of the information processing takes place. With so much of the brain hidden, researchers have a hard time seeing exactly which parts of the cortex are doing what and how they are related to one another. A Mercator-like flat map of the brain can be viewed in three ways:
• Euclidean, which is flat like a road map. Distance is measured or scaled as expected.
• Hyperbolic, which is disk-shaped and allows the map focus to be changed so that the chosen center is in sharp focus and the edges distorted, much like moving a magnifying glass over a piece of paper.
• Spherical, which wraps a flattened brain image around a sphere.

The same type of reasoning is used to try and create a map of the structure of the internet:

Hyperbolic map of the internet

### Managing Data Using Higher Dimensions

The complexity of higher dimensions can be experienced regularly in our data driven society. Any time we measure more than 3 variables for a poll, we are inside of a higher dimensional space.
 For example, here experts collect data for a large sample of individuals and assigned the participants in one of two classes. 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 (which we'll see later on this semester), but instead of finding a "best fit" line to all of the data, we find the higher dimensional 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

• Find a partner and choose one computer to click on the links. One of you should read this page as the other follows the directions. View the numerical data that was actually used in the heart disease analysis (description of the data). Using Select All and then Copy under Edit, copy the numerical data from this link. Open up Word (blue W icon) and paste the data into Word. Under Edit, Replace all of the instances of , with ^t (which is the code for a tab formatting that Excel can read). Then under Edit, Select All and then Copy. Open up Excel (green X icon) 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.

• Question 4 Each column is a different dimensions worth of data. How many dimensions is this space (ie how many columns)?
• Question 5 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. You may now quit Excel without saving your file.

### What is the 4th Physical Dimension?

We have heard that physicists think that the universe has many more physical dimensions than we directly experience. We can try and understand the 4th physical dimension by thinking about how a 2D Marge can understand the 3rd physical dimension. For example, when Homer disappears behind the bookcase, or when she sees shadows of a rotating cube, she experiences behavior that does not seem to make sense to her. In fact, since it is 3D behavior, it does not make sense in 2D. But, it is in this indirect way that 2D Marge can gain an appreciation for 3D.

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, physicists and mathematicians assert that 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.

• In order to try and gain some understand for more physical dimensions, we'll examine images and movies from Davide Cervone's talk on The Cube and the Hypercube: Rotations and Slices

First look at Dimensional Connections and Into the Fourth Dimension

• Question 6 How is a hypercube formed from a cube? Use an analogy similar to Professor Frink's description of how a cube is formed from a square.

Next examine Counting Boundaries, Hypercube Boundary, and the movie The Eight Cubical Faces of a Hypercube - to play the movie, use the .

• Question 7 How many 3D box boundary pieces (which Davide Cervone calls "faces of a hypercube") does a hypercube have? Experts think a hypercube is not one of the possible shapes of our universe because it has edges to fall off of.

Now examine A Cube Unfolded and A Hypercube Unfolded.

• Question 8 Use a websearch to determine what famous artist used the unfolded hypercube in his 1954 painting?

Explore the movies Rotating a Cube and Rotating a Hypercube by using the . If it is viewed properly, the second movie should stretch the limits of your imagination.

Artist Toni Robbin, from the Life by the Numbers NOVA video, creates shadows of the hypercube on canvas. Here is a recent (2007-4 56" x 70", Coll: the artist) work of his: .

He also mentions that he finds it a privilege to live in a time where advanced mathematics is represented in pictures instead of only equations.

• Question 9 What might one layer of Homer's skin look like in 4D if he were to change from 3D to 4D? (Hint: A layer of skin looks like a 2D piece of paper with holes or pores in it - think about what familiar food this might resemble if it gained a dimension/ie gained thickness.)

### The Geometry of our Universe

The shape of space is directly related to whether the space is Euclidean, spherical or hyperbolic. Mathematicians are working with astronomers and physicists in order to try to solve this problem. Greek mathematicians were able to determine that the earth was round without ever leaving it. We hope to answer the most basic question about our universe in a similar manner.

Some cosmologists expect the universe to be finite, curving back around on itself. 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 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."

### 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 that 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.

 Futurama TM and copyright Twentieth Century Fox and its related companies. This educational use is not specifically authorized by Fox. 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. We saw one example like this in the Futurama episode I, Roommate when Fry, Leela, and Bender were looking at an apartment resembling Escher's 1953 Relativity work.

 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. Use the reading above to help you answer these questions:

• Question 10 Describe how a Euclidean 3-torus can be formed from a cube or box by explaining how you would glue the faces of a cube together. This is one of the possibilities for a finite universe without any edges, since we glue the edges together, and many 3-D video games (like Valve's Portal) make use of these types of universes.
• Question 11 Each Euclidean 3-torus below has a familiar solid drawn in it once you perform you gluing instructions from the last question. For example, the fourth picture generates a cone because the top and bottom face gluing yields 2 half cones, and the left and right faces are glued together forming 1 full cone. Give the name of each of the other surfaces.
• Question 12 If our universe were a quarter-turn Euclidean space, we might be able to tell by looking for repeated patterns of stars in different directions which would differ by the same angle used to identify the faces - what angle is this? Hint: What is the angle in a full turn? What is the angle in one-quarter of a turn?
• Question 13 There are 9 other shapes that we can obtain as possibilities for a Euclidean universe via gluing with twists and turns. Unmarked walls are glued to one another in the simple, straight-across way while the marked side shows whether to glue straight across, with a reflection or a rotation by identifying corresponding squares (squares that are filled in the same get glued together). For example, the second space is the Klein space because it is identified via a vertical mirror plane of reflection that cuts the cube in half. For the remaining three spaces, write down which space is the 3-torus, quarter-turn space, and half-turn space.

• ### Spherical Universes

Einstein based his theory of relativity on a higher dimensional sphere, but there are other ways to make spherical universes.
 Bender's Big Score: Greenwaldian Theorem A spherical dodecahedron from Paul Nylander

• Question 14 Go to the dodecahedron dice table. Notice that if you want to glue opposite faces together, we will need to do so with a twist, since the straight across gluing won't work. How many faces does it have?
• Question 15 Try to fit them together to tile our space. Can the dodecahedral dice be placed together without any space remaining? If so, it will form a Euclidean space, like the 3-torus, that satisfies the Pythagorean theorem. If not, the space won't satisfy the laws from high school, and we'll have to distort the sides, like Escher, by either bowing them out (spherical) or in (hyperbolic) to make them fit. Did they fit together perfectly?

In fact, in the same way as 12 spherical pentagons fit together perfectly to make a sphere, which is the surface of a 3-D ball, 120 spherical dodecahedrons fit together perfectly to make a hypersphere, and so we can identify opposite faces together to make a finite spherical universe without any edges.

### Hyperbolic Universes

 By gluing together corresponding sides of a distorted version of this 18-sided figure (for example, the pentagon faces get glued together), we obtain a hyperbolic universe. Mathematical objects are sometimes, but not always, named after the first person to discover them. Just as Pythagoras was not the first to come up with the Pythagorean theorem [which goes back to Babylonian times], I was not the first to come up with the spherical version now known as the Greenwaldian theorem in the Futurama universe [which probably goes back to Menelaus of Alexandria]. However, this space known as the Weeks manifold, after Jeff Weeks, was first discovered by him.

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

• Question 16 Search the web to find information on the Weeks manifold. Write down one related item.

A hyperbolic dodecahedron from Paul Nylander

• For tomorrow, go through the study guide for the test and message me or write down at least one question or comment for a review session during class. Recall that the advice from last semester's students was to know everything on the study guide. In addition take a try of ASULearn Material Review Quiz [participation requires at least one try of the quiz, but the specific grade does not matter.]

We will go over the lab in class tomorrow. If you are finished, you may leave, or stay to ask me questions in office hours.

### References -- Adapted from excerpts taken from:

• Davide Cervone's materials
• Cathy Gorini Geometry at Work
• David Henderson Experiencing Geometry in the Euclidean, Spherical, and Hyperbolic Spaces
• Jean-Pierre Luminet, Glenn D. Starkman and Jeffrey R. Weeks Is Space Finite?, Scientific American
• Diane Martindale Road Map for the Mind: OLD MATHEMATICAL THEOREMS UNFOLD THE HUMAN BRAIN, Scientific American
• Jeff Weeks Exploring the Shape of Space
Dr. Sarah J. Greenwald, Appalachian State University

Futurama TM and copyright Twentieth Century Fox and its related companies. This web site is for educational use only. This web site, its operators, and any content contained on this site relating to Futurama are not specifically authorized by Fox.