Good Friday

(This is a copy if a previous post, but appropriate for today).

Courtesy of the Association of Christians in the Mathematical Sciences:

Salvador Dali’s painting Corpus Hypercubus (1954) is a fascinating visual representation of a mathematical metaphor for the theological mystery of crucifixion.

Corpus Hypercubus, Salvador Dali (1954)

Many people are familiar with how to unfold a cube from three dimensions into two as shown in the figures below. Some of the edges of the cube are separated so that the resulting collection of squares can be unfolded into a planar shape. The resulting diagram is called a “net” for the cube. The net is not unique but one common net resembles a cross.

Dali painted the cross in Corpus Hypercubus as a hypercube unfolded into 3-dimensional space. The hypercube consists of eight three-dimensional cubes for hyperfaces. Each hyperface is attached along a two-dimensional square face to six of the other hyperfaces. Just as one can unfold a cube, one can also unfold a hypercube into the shape depicted in Dali’s painting.

Using the analogy of a (mysterious) higher-dimensional object unfolded into three dimensions, Dali depicts the theological mystery of the crucifixion as an event that originated in a higher plane of existence and then unfolded into the world that we perceive. With this understanding, Corpus Hypercubus communicates the idea that though one can discuss the necessity of the Jesus’ sacrifice for salvation or study theological ramifications of the cross, one can only do so by analogy because human nature simply cannot perceive the scope of God’s plan.

More from the ACMS:

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Flatland: A Project of Many Dimensions

Last fall my advanced (freshmen) geometry classes completed a semester project on Flatland, a book by Edwin A. Abbott. In case you haven’t heard of this obscure yet significant work, let me start this post by providing a brief overview of the book and then I’ll describe the structure of the project.

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The Book (From Amazon.com):

Flatland is one of the very few novels about math and philosophy that can appeal to almost any layperson. Published in 1880, this short fantasy takes us to a completely flat world of two physical dimensions where all the inhabitants are geometric shapes, and who think the planar world of length and width that they know is all there is. But one inhabitant discovers the existence of a third physical dimension, enabling him to finally grasp the concept of a fourth dimension. Watching our Flatland narrator, we begin to get an idea of the limitations of our own assumptions about reality, and we start to learn how to think about the confusing problem of higher dimensions. The book is also quite a funny satire on society and class distinctions of Victorian England.

The Project:

The essence of this project was to have students interact with the subject matter of this book through weekly reading comprehension quizzes, two in depth class discussions, and a culminating extension project done in response their understanding of the book.

Reading Schedule (links are to the reading comprehension quizzes):

Class Discussion:

There were two major class discussions: one over the first half of the book that extended to a discussion of Flatland and society, and one over the second half of the book that extended to Flatland and theology. One week prior to our class discussion the students were given the following prompts and extra reading resources.

Discussion 1: Flatland and Society: In addition to being an interesting piece of Geometric fiction, the first half of Flatland serves as a satirical social commentary on Victorian England. Do a little research on what society was like at during that period. Be prepared to discuss this social culture and link it by analogy to the society of Flatland.

In your preparation I would like for you to wrestle with the following quote about mathematical advances in Victorian England (taken from Equations from God: Pure Mathematics and Victorian Faith, by Daniel Cohen) where we find “origins of the rebirth of abstract mathematics in the intellectual quest to rise above common existence and touch the mind of the deity” (emphasis added). Relate this comment to the notions presented in Flatland.

Here are some questions that you might think of answering prior to class discussion, and these concepts may help you begin to organize your thoughts. It should be noted that none of these questions have easy answers, but they do have plausible ones. I am not looking for you to give a “right” answer, but to wrestle through the concept and tell me what you think.

  1.  How does the society of Victorian England mirror that of Flatland?
  2. How are women depicted in Flatland? How were they treated in Victorian England? What point is the author trying to make in his depiction of women?
  3. Why might Flatlanders be interested in rising above common existence? Why would this be said of people in Victorian England?
  4. How does A. Square “touch the mind of deity?” How might we “touch the mind of deity” as we study mathematics?

Resources:

Discussion 2: Flatland and Theology: Edwin A. Abbott was not only trained in mathematics but theology as well. The second half of the book can be seen as a theological analogy: the visitor from Spaceland interacts with Flatlanders in a way that might mirror how God, who is not bound by the limits of three dimensions, interacts with our three-dimensional world. Draw out this analogy using specific examples from the book and from the Bible to answer the question: “How do we experience God?” Be sure to cite any references you make to Flatland and use biblical examples of God’s interactions with His creation to support your ideas.

Here are some questions that you might think of answering prior to class discussion, and these concepts may help you begin to organize your thoughts. It should be noted that none of these questions have easy answers, but they do have plausible ones. I am not looking for you to give a “right” answer, but to wrestle through the concept and tell me what you think.

  1. How might our universe appear to a being outside of it? How did Flatland appear to the Sphere?
  2. What does it mean for God to be “over all, through all, and in all” (Ephesians 4:4-6)? How might this be said of the Sphere in relation to Flatland?
  3. How might it be possible for God to be omniscient (know what is happening in all parts of the world at the same time)? How can this be said of the Sphere in relation to Flatland or of the Square in relation to Lineland or Pointland?
  4. What were A. Square’s limitations in fully understanding the Sphere? What are our limitations in fully understanding God?
  5. What would we expect to see if/when God entered the world? How might it be similar or different from when the Sphere entered Flatland?

Resources:

Project Options:

Choose from one of the following options as a project response to your reading of Flatland. The resources cited for each option are available on the course webpage under the “Projects” folder. You are not restricted to these resources, in fact you are encouraged to find additional ones, however the cited resources must be analyzed prior to beginning your project.

The project proposal needs to include your name, class period, and the project option you have selected. You also need to provide a preliminary outline or plan of attack. The big idea here is that you need to demonstrate that you have spent time considering your project and have a clear plan for moving forward.

All papers, while they may deal with a subject that is not explicitly mathematical, need to contain significant geometric content. Show me that you understand the geometry as it is presented in the book and then take that knowledge to analyze the subject matter of your paper. Papers will be graded more on mathematical understanding than on writing ability, though proper grammar and spelling need to be used (see rubric for more details).

Option 1: Book review, extra reading plus a 2 page review. Read one of the following “sequels” to Flatland and write a short review:

  • Sphereland: A Fantasy about Curved Spaces and an Expanding Universe, by Dionys Burger (ISBN: 0064635740)
  • Flatterland: Like Flatland, Only More So, by Ian Stewart (ISBN: 073820675X)

The book review should take 1 page to summarize the book and 1 page to critique it. In the critique you need to address what the author did well/poorly and why. You should also address what impact the book had on your understanding of Flatland. Be sure to comment on significant/interesting mathematical descriptions as they are presented. You will find several sample book reviews linked on the course website to give you a feel for what a book review looks and sounds like.

Resource: How_to_write_a_book_review

Option 2: Creative writing assignment, minimum of 4 pages, on one of the following topics :

  • Rewrite A. Square’s description of the visitor from Spaceland had that visitor been a Cube instead of a Sphere. Then rewrite the experience again if the visitor had been each of the other Platonic Solids (Tetrahedron, Octahedron, Dodecahedron, Icosahedron). This amounts to writing ¾ to 1 page on each solid.

Resource: Descriptions of Platonic Solids

  • Write an additional chapter for the book in which the Sphere is visited by a person/object from the 4th dimension. Write the Sphere’s description of the visitor in a way that mirrors A. Square’s description of his encounter with the Sphere.

Resources: Short story “The 4D Doodler” by Graph Waldeyer & “What is the 4th dimension?” by Eric Saltsman

Option 3: Explore 4 Dimensional Hypercubes: Complete questions 1-4 on the Hypercube Activity posted on the class website (Kuyers Institute Lesson on Hypercubes). This project will culminate with an activity in which you build a 3D model of a 4D hypercube.

Resources: Helpful websites are linked within the Hypercube Worksheet document & “What is the 4th dimension?” by Eric Saltsman

Project Rubric

Additional Resources: