The purpose of this test experience is to help you understand the material and make connections. Research has shown that the effort you expend in studying for tests and clearly explaining your work in writing solidifies your learning.

### At the Exam

• One 8.5*11 sheet with writing on both sides allowed. You can put anything you like on your sheet.
• You may bring your child's ball with you, with the writing on it from the earth intro activity.
• One scientific calculator or graphing calculator allowed (but no cell phone nor other calculators bundled in combination with additional technologies)
• You may have out food, hydration, ear plugs, or similar if they will help you (however any ear plugs must be standalone--no cell phone, internet or other technological connections)
• Your grade will be based on the quality and depth of your responses in the timed environment (the test must be turned in when time is called).
• You will be expected to answer questions about any of the various items from class, lab, and on ASULearn. The test will be a mixture of computational questions related to geometric equations, directed questions related to geometric spaces, short answer or short essay questions on our activities, and questions on the "big picture" themes and more. There is NO need to write in full sentences - bullet points, etc. will be fine.

Review
• Here is a partial sample test
and some partial answers so that you can see and have some practice with some diverse examples of the formatting and style of questions. The actual test will differ.
• Review class and ASULearn activities and review problems as the vast majority of exam questions are taken from there.
• Here is an overview of topics. Be sure you could respond to questions on these. I want you to understand the material and I am happy to help!

### Computational Questions Related to Geometric Equations and Spaces

Eratosthenes
• Compute the circumference of the earth from local data

Angle sum, straight feeling paths and parallels
• You might be presented with one of Escher's works, asked to calculate the sum of the angles in a triangle in the space represented, and specify whether the space is then Euclidean, spherical or hyperbolic. video review of Escher. Embedded in this analysis is the knowledge that the sum of the angles in a triangle is important since it determines the geometry of a space: 180 degrees for flat Euclidean spaces, greater than 180 degrees for spherical spaces, and less than 180 degrees for hyperbolic spaces.
• Describe an argument of why the angle sum is 180 degrees in Euclidean (folding using parallels, or walking/driving the angle sum) and why it is smaller in hyperbolic and larger in spherical geometry
• Be able to plug into a spherical, hyperbolic or flat angle sum relationship to solve for the sum or an angle, like A + 90 + 45 > 180 in spherical geometry so A > 45.
• Be able to analyze straight feeling/symmetric (to a bug) paths and parallels (or a lack of parallels) in an Escher representation, a perspective drawing, or on the earth, and count the number of parallels (0 for spherical, 1 for Euclidean, multiple for hyperbolic)

Pythagorean theorem
• Be able to plug into a spherical, hyperbolic or flat Pythagorean theorem to determine (for instance) problems like the fact that in hyperbolic geometry, the hyperbolic hypotenuse would be longer than a2 + b2 because a2 + b2 < (hyperbolic c)2.
• Describe a geometric argument of why the Pythagorean theorem is true in Euclidean geometry (water wheel) and false in spherical geometry (string argument) and has the opposite relationship as hyperbolic.
• Be able to algebraically demonstrate the Pythagorean theorem based on the picture in the Zhou Bi Suan Jing or Chou Pei Suan Ching which had a large square broken up into 4 triangles and a smaller square

Inverse square law
• Inverse square law for light: Brightness changes as 1/ the square of the distance for Euclidean, with a < relationship for hyperbolic (i.e. dimmer at a given distance than in Euclidean) and a > relationship for spherical (i.e. brighter at a given distance than in flat space). This was used in the analysis of the supernova experiments.

### Directed Questions Related to Geometric Spaces

Wraparound spaces
• Klein bottle tic-tac-toe: You might be given a partially completed Klein bottle tic-tac-toe game and asked to mark a winning move. You may wish to review by playing a few games. In addition, you should understand the global tiling view - where identified moves are marked above, below, to the right, and to the left of the board, as on the spaceship picture. Here is a video review of Klein bottle tic-tac-toe.

Earth and Universe
• Review class activities on the earth and universe For example, you should be able to explain in depth the various ways we explored shortest distance paths on a sphere (the car, masking tape, string, Chicago-Rome, symmetry...) and the ways we explored the Pythagorean theorem on the sphere (string, algebra, computer program on a transparency, Futurama Greenwaldian theorem), along with more general questions like describing one of the possible finite shapes without edges for our universe, or providing responses for some of the other questions.

Cross-sectional views
• Be able to address what a carp, Arthur Square, or 2D Marge would see if an orange passed by them (a sequence of lines/curves that appear, get bigger and disappear), as well as what the full cross sections would be (circles that get bigger, then smaller, then disappear). Instead of an orange, think about what what happens of spaces with 1, or 2, or more holes pass by them, like a mug or a 2-holed donut.

Analyzing experiments to determine the geometry of the universe
• You might be asked to describe in detail some of the methods being used to attempt to discover the shape of the universe, as well as our classroom activities on those critiques (Gauss' triangle but light bending with gravity, measuring too small a triangle, and other problems, Rob Kirshner's Supernovae distance/brightness experiments but experimental error, and Supernovae not necessarily exploding at the same brightness, looking for repeated patterns like the quarter-turn space but difficulty recognizing the patterns and looking at convenience sampling, density equation data but difficulty agreeing on the meaning of the data...)

Folding/Gluing spaces
• You should be able to identify spaces from a region with its gluings, like we have seen in activities on 2D universes and the universe.
• You should be able to explain the region we used and the gluings or creation of a cube, 2-holed donut, Klein bottle, hypercube, 3-torus
• You should be able to examine whether there is any space leftover when we put octagons or dodecahedrons together (there is, and we bowed-in to put more octagons in and create the hyperbolic 2-holed donut from them, and we bowed-out to fill up the extra space and create the spherical dodecahedral space)
• You should also be able to identify a familiar 2D surface drawn in a 3-torus it once you perform the gluings of faces directly across.

### "Big Picture" Themes and More

Real-life applications
• You might be asked about real-life applications of some of the material we have been studying, like a dodecahedron, hyperbolic geometry, a hypersphere, higher spatial dimensions or more. A good review for this would be the review practice , but the question wouldn't be in a multiple choice format, instead I would ask a real-life application of the geometric object or concept. You should also know how to describe the spaces themselves.

Changes in world view
• You might be asked about some of the changes in world view that came with mathematical discoveries, from the videos and homework readings. A good review for this would be the review questions, but again it wouldn't be in multiple choice format.

Thematic questions
• You might be asked about thematic issues, like an example of local to global issues and how they related to Eratosthenes, the sum of the angles, the Pythagorean theorem, or people around the world, a reflection from this segment on what is mathematics/what is geometry, what mathematics has to offer and how it is useful, the diverse ways that people succeed in and impact mathematics, or the themes of what a space looks like, how do we know (the theme of truth--When are we convinced? What are the consequences of certain truths? What is the role of chance and probability?) and how do we represent it. For instance, a question could be phrased like: Name and discuss an instance from the geometry segment where probability and chance played a role, or a more directed question, like analyze the role of probability and chance in computing the density and geometry of the universe, how .4% related, how 1 in 3000 chance related, or how 1/180th of a degree related. Similar types of open ended and/or directed questions could ask about the other themes.