The Geometry of Our Universe Lab

We'll continue to engage with theories about the geometry and shape of our universe as well as connections to art, philosophy, physics, medicine, 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 with an open mind, then these ideas should stretch the limits of your imagination. In order to help you, I have pulled together readings, hands on activities and movies.

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 (when needed) one person reading these directions from their computer as the other performs the activities, and taking some notes on the readings in addition to answering the questions.

The Geometry of our Universe

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.

Euclidean Universes

NASA states that the satellite probe named "the universe is flat with only a 0.4% margin of error". Flat universes satisfy the laws of Euclidean geometry from high school. One possibility is an infinite Euclidean space. Another is a finite space with any edges glued together, like those mentioned in the homework readings. NASA also states: "since the Universe has a finite age, we can only observe a finite volume of the Universe. All we can truly conclude is that the Universe is much larger than the volume we can directly observe."

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 finite wraparound Euclidean universe and they all arise from a cubical block of space with the faces glued to together.
Work with a partner or two and answer these questions together:

  • Question 1
    Have one person read these instructions while the other follows them.
    Part a)
        *Download Torus Games for Mac OS 10.12 or later. You can also download this free software onto other devices from the above link, if you prefer.
        *Select 3D Maze.
    Rotate the maze around and drag the sphere through a face. Don't worry about finishing the maze, but do play enough so that you get the idea of the gluings. This is one of the possibilities for a finite universe without any edges (since we glue them together), and many 3-D video games (like Valve's Portal) also make use of these types of universes.

    Part b) Give the gluing instructions that describe how this Euclidean 3-torus can be formed from a cube or box by explaining how you would match up the faces of a cube together [which faces get glued and is it straight across, via a reflection, or in some other way?]. Note there are 3 different sets of face gluings.

  • Question 2 There are 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. The half-turn space is a 180 degree rotation of the front to get to the back. For the remaining two spaces, write down
    Part a) which space is the 3-torus and
    Part b) which space is the quarter-turn space (like in the homework reading).

    Klein space

    half-turn space

  • Question 3 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 4 Each Euclidean 3-torus below has a familiar solid drawn in it once you perform you gluing (just like in the homework reading). 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.

    Sketch the following in your notes and give the name of each of the other surfaces. Hint: cube, sphere

  • Spherical Universes

    Historically, 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. 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 edges to fall off of. One might still ask what is outside the universe. but this question supposes that the ultimate physical reality must be that the space sits inside of something else. 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."

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

  • Question 5 Go to the dodecahedron dice table and pick up 3 dodecahedrons of the same size. Notice from looking at just one dodecahedron that if you want to glue opposite faces together, we will need to do so with a twist, since the straight across gluing will not work (for example, a pentagon face must be rotated to match up with the one opposite to it).

    How many faces does the dodecahedron have?

  • Question 6 Once they are glued together, a person in the space would see a tiling view. Try to fit them together along an edge (one of the line segments between faces). If 3 dice fits perfectly, it will form a Euclidean space (like cubes fitting together to make a 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 the flat faces of 3 dodecahedron fit together perfectly along a line segment edge, leaving no space in excess?

    Return the dice.

    In fact, in the same way as 12 spherical pentagons fit together perfectly to make a spherical dodecahedron, as in the picture in the above middle, then 120 spherical dodecahedrons fit together perfectly to make a hypersphere (above right). So we can identify opposite faces of the spherical dodecahedron together to make a finite spherical universe without any edges. This is just one of infinitely many spherical possibilities (which have been classified completely). Higher dimensional spheres have been used in protein modeling in biology, artificial intelligence and machine learning, and statistics.
  • Hyperbolic Universes

    By gluing together corresponding sides of a bowed-in hyperbolic 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 7 Search the web to find information on the Weeks manifold. Write down one related item.

    A hyperbolic dodecahedron from Paul Nylander

    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. Mercury's orbit about the Sun is slightly more accurately predicted when Hyperbolic Geometry is used in place of Euclidean Geometry and hyperbolic models, and hyperbolic crystal structures can store more hydrogen or absorb more toxic metals. In addition, artists and mathematicians have been creating hyperbolic coral reefs, maps of the brain to diagnose or monitor neurological diseases, and ven of the internet to try and reduce the loads on routers.

    Hyperbolic reef                                                           Hyperbolic map of the internet

  • Question 8 Which real-life application of hyperbolic geometry do you find most compelling?

    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.

    Heart Disease

  • Download this data file and open it in Excel (green X icon) if it doesn't automatically open.

  • Question 9 Each column is a different dimensions worth of data. How many dimensions is this space (ie how many columns)?
  • Question 10 Each patient is a different row. How many patients were studied? Hint: subtract 1 from the rows used, since the first row contains the data descriptors rather than a person.

    This data was the real data that was used to study patients in this higher dimensional data space (description of the data). Data was collected from patients in the US, Hungary and Switzerland. New patients have since been diagnosed inside of higher dimensions using a higher dimensional separating plane:
    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. 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.

  • Question 11 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 dairy product this might resemble if it gained a dimension/ie gained thickness.)

    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

  • Question 12 First look at Into the Fourth Dimension. How is a hypercube formed from a cube? Write down the analogy (and notice that it is similar to Professor Frink's description of how a cube is formed from a square from the 2-D universes lab).

  • Question 13 Next examine the movie The Cubical Faces of a Hypercube - to play the movie, use the . How many cubical faces (boundary pieces) 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.

  • Question 14 Now look at Hypercube Boundary to look for a pattern between the dimension and the number of boundary pieces. If nD is the dimension, how many boundary pieces should it have?

  • Question 15 Now examine A Cube Unfolded and A Hypercube Unfolded. Use a web search 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 Tony Robbin, from the Life by the Numbers NOVA video, creates shadows of higher dimensions on canvas. Here is a recent work of his (Oil on Canvas, 56 by 70 inches):

    I find it a privilege to live in a time where advanced mathematics is represented in pictures instead of only equations.
    [Tony Robbin, Life by the Numbers]

  • Question 16 Search the web for: Tony Robbin hypercube and write down one related item.

  • Read over the ASULearn Glossary/Wiki entry on truth and consequences in geometry.
  • Show me your lab responses before you leave and ask me any questions on this or the upcoming exam.

  • 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

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