Author:
Amanda Hager, Gwendolyn Morel
Subject:
Mathematics
Material Type:
Lesson
Level:
Academic Lower Division
Provider:
University of Texas at Austin
Tags:
  • D2S2
  • Mindset
  • License:
    Creative Commons Attribution
    Language:
    English
    Media Formats:
    Text/HTML, Video

    Education Standards

    3.6 4D Cubes and Triangles

    3.6 4D Cubes and Triangles

    Overview

    TCCNS CourseMATH 1332: Contemporary Mathematics
    UT Austin CourseM 302: Introduction to Mathematics

    Introduction

    Suggested Resources and Preparation

    Materials and Technology

    • For the instructor: Manipulatives for the boundaries of the 4D simplex and cube are of great value. Business cards can be used to fold a hollow paper model of the tesseract boundary, or you could glue together dice or foam cubes to create models. Large tetrahedra are hard to come by, but you can use four-side dice or fold them origami-style.

    • Manipulatives for assembling boundaries of 3D objects are also handy. Empty cardboard boxes can be moved about and labeled to show how they can be assembled. Boundaries of tetrahedra and other polyhedra could be cut out with scissors and moved about. 

    Prerequisite Assumptions

    It is expected that students are proficient with the contents of Section 3.4 and Section 3.5 of this course.

    Overview and Student Objectives

    Lesson Length

    50 minutes

    Lesson Objectives

    Students will understand that:

    • In math there are families of objects that are all built in similar ways, one object in each dimension. Two such families are cubes and simplices (or triangles).
    • One way we can try to visualize four-dimensional objects is by analyzing their boundary, which when sliced apart can be drawn in 3D space.

    Students will be able to:

    • Count combinatorial features (number of vertices, edges, faces, fillings, etc.) of cubes and simplices/triangles of dimension 0-4.
    • Describe or draw how a paper cutout could be taped together to form the boundary of a 3D object.
    • Describe or draw how 3D solids could be glued together in 4D space to create the boundary of a 4D object.

    4D Cubes and Triangles

    Understanding an object via its boundary

    As a way of proving you learned some things about 4D, I'm asking you to understand two 4D objects a lot: a 4D cube and a 4D triangle (or simplex). For this section, we're going to be taking what's called a combinatorial approach. We're going to think of objects as being built out of lower-dimensional objects as if they were building blocks. A lot of our work then becomes counting how many of each kind of object there are, and discussing how those building blocks are put together.

    To get started, remember that right before the last exam I had you learn about boundaries of objects.

    Terms

    • The boundary of an object are all the points in the object that are adjacent to parts of the object and points that are not in the object. You can approach these points of the object both from outside and inside the object.

    Family of cubes

    To start with, we will define a 0D cube to be a single vertex, and we will define a 1D cube to be a line segment (which has a boundary of two vertices).

    Now imagine building a square out of 0D and 1D pieces. Think of 4 edges and 4 vertices, all joined together in a square shape, so it makes a wire square. Then you shade in the middle with 2D area.

    left: hollow square, vertices in the four corners and four edges. Center: a right pointing arrow. Right, a shaded square with four vertices in the corners and four edges.

     

    Here are some facts:

    • For short I will say "A 2D square is made up of 4 vertices, 4 edges, and one face."
    • At each vertex we can see that two edges meet up. Using the formulas from the last unit, we could say "the number of vertices will be equal to \(\frac{4\cdot2}{2}=4\) because there are four edges with two vertices each, and we are joining together two vertices at a time."

    Next, think of building a 3D cube by essentially building an empty box and then injecting the 3D volume afterward.

    Left: hollow cube with 8 vertices in corners and 12 edges. Center: right facing arrow. Right: solid cube with same vertices and edges emphasized.

     

    Here are some facts:

    • For short I will say "The 3D cube is made up of 8 vertices, 12 edges, 6 faces, and one filling." 
    • At each vertex we can see that three edges meet up. Using the formulas from the last unit, we could say "the number of vertices will be equal to \(\frac{6⋅4}{3}=8\) because there are six square faces with four vertices each, and we are joining together three vertices at a time."

    This video is a big one. An entire lecture on how you can hold the 3D boundary of a 4D cube in your hands and squint your eyes real hard and use a Sharpie and try to imagine the shape of the 4D cube.

    In other words, here is the partly disassembled 3D boundary of a 4D cube.

    Set of eight 3D cubes connected together. Four are arranged in a column, and then four more are joined to the exposed faces of the second cube from the top.

     

    Here are some facts:

    • For short, I will say the 4D cube is made up of 16 vertices, 32 edges, 24 faces, 8 3D fillings, and 1 4D "hyperfilling."
    • There are 8 cubes in the boundary, we are gluing together two faces at a time, there are three cubes joining together around every edge, and there are four cubes joining together around every vertex. So using the formulas from the last unit, there will be \(\frac{8⋅8}{4}=16\) vertices, \(\frac{8⋅12}{3}=32\) edges, and \(\frac{8⋅6}{2}=24\) faces.
    • To finish assembling the boundary in the picture, you have to leave 3D world and enter the fourth dimension. Only then can you finish gluing pairs of 2D faces together. Then you inject the object with "hypervolume" and now you have a 4D cube.

    Family of triangles

    Did you notice that you could build a 1D cube using 2 0D cubes, a 2D cube using 4 1D cubes, and a 3D cube using 6 2D cubes, etc? So in general, you can build an n dimensional cube using 2n cubes of one dimension lower.

    I can say a similar thing about triangles.

    • A 0D triangle is a single vertex.
    • A 1D triangle is made from 2 0D triangles and then you color in an edge.
    • A 2D triangle is made from 3 1D triangles and then you shade in the area.
    • A 3D triangle is made from 4 2D triangles and then you fill in the volume (this is a tetrahedron).
    • A 4D triangle is made from 5 3D triangles and then you fill in the hypervolume.
    • An n dimensional triangle is made from n+1 triangles of one dimension lower and then you fill in the hypervolume.

    This video shows you some visuals on how that works and how you can count the features.

    Bonus! Art and Math!

    It took a while for the math world to figure out how to talk about higher dimensions. It started in the 1700s, slowly, and was formally and precisely defined by Riemann in the mid 1800s. However, the concept of 4D was popularized by a British mathematician named Hinton in 1880 in an essay titled "What is the fourth dimension?" From there, the concept trended. The Flatland novel was published in 1884, and artists such as Max WeberEtienne-Jules Marey, and Marcel Duchamp experimented with ways of depicting the fourth dimension on canvas or film. But one of my favorites might be Dalí..

    Dali_Crucifixion_hypercube.jpg

    Crucifixion (Corpus Hypercubus), Salvador Dalí, 1954.

    Quiz Questions

    Question 1

    The boundary of a 1D triangle is 2 vertices (also known as 0D triangles). The boundary of a 2D triangle is 3 line segments (also known as 1D triangles).

    1. How many 2D triangles are there in the boundary of a 3D triangle (tetrahedron)? 4
    2. How many 3D triangles are there in the boundary of a 4D triangle? 5
    3. How many 4D triangles are there in the boundary of a 5D triangle? 6

    Question 2

    The boundary of a 1D cube is 2 vertices (also known as 0D cubes). The boundary of a 2D cube (also known as a square) is 4 line segments (also known as 1D cubes).

    1. How many 2D cubes are there in the boundary of a 3D cube? 6
    2. How many 3D cubes are there in the boundary of a 4D cube? 8
    3. How many 4D cubes are there in the boundary of a 5D cube? 10

    Question 3

    Complete this chart of features of cubes in various dimensions.

    DimensionsVerticesEdgesFaces3D fillings4D "fillings"
    010000
    121000
    244100
    3812610
    416322481

    Question 4

    Complete this chart of features of triangles in various dimensions.

    DimensionsVerticesEdgesFaces3D fillings4D "fillings"
    010000
    121000
    233100
    346410
    45101051

    Question 1

    The boundary of a 1D triangle is 2 vertices (also known as 0D triangles). The boundary of a 2D triangle is 3 line segments (also known as 1D triangles).

    1. How many 2D triangles are there in the boundary of a 3D triangle (tetrahedron)? 
    2. How many 3D triangles are there in the boundary of a 4D triangle? 
    3. How many 4D triangles are there in the boundary of a 5D triangle? 

    Question 2

    The boundary of a 1D cube is 2 vertices (also known as 0D cubes). The boundary of a 2D cube (also known as a square) is 4 line segments (also known as 1D cubes).

    1. How many 2D cubes are there in the boundary of a 3D cube? 
    2. How many 3D cubes are there in the boundary of a 4D cube? 
    3. How many 4D cubes are there in the boundary of a 5D cube? 

    Question 3

    Complete this chart of features of cubes in various dimensions.

    DimensionsVerticesEdgesFaces3D fillings4D "fillings"
    0     
    1     
    2     
    3     
    4     

    Question 4

    Complete this chart of features of triangles in various dimensions.

    DimensionsVerticesEdgesFaces3D fillings4D "fillings"
    0     
    1     
    2     
    3     
    4     

    Homework Questions

    Question 3.6.1

    1=1, 2-1=1, 4-4+1=1, 8-12+6-1=1, 16-32+24-8+1=1.

    Question 3.6.2

    1=1, 2-1=1, 3-3+1, 4-6+4-1=1, 5-10+10-5+1=1.

    Question 3.6.3

    Identify a piece of the knot to "lift into the fourth dimension" by circling it. Then lift it, which means you draw a second drawing and just don't draw that piece. Then you switch which string is on top and lower it back down, which means a third drawing where the crossing is switched. Then (important), you show how to untangle it with words or drawings.

    Question 3.6.4

    Every time you see 8 vertices, 12 edges, and 6 faces that are all connected in the same pattern as the normal cube, you get a cube. There are 8 visible in the picture.

    Question 3.6.5

    In a 1D cube, max distance is 1. In a 3D cube, max distance is 3. In a 4D cube, max distance is 4. In an n dimensional cube, max distance is n. Drawings work for showing the 1D and 3D examples.

     

    Question 3.6.1

    For the 0D, 1D, 2D, 3D, and 4D cubes, compute the Euler characteristic. The formula for this is to take the number of 0D pieces, subtract 1D pieces, add 2D pieces, subtract 3D pieces, and add 4D pieces (and so on for higher dimensions). This is a generalization of the V-E+F formula from the last unit.

    Question 3.6.2

    For the 0D, 1D, 2D, 3D, and 4D triangles, compute the Euler characteristic. The formula for this is to take the number of 0D pieces, subtract 1D pieces, add 2D pieces, subtract 3D pieces, and add 4D pieces (and so on for higher dimensions). This is a generalization of the V-E+F formula from the last unit.

    Question 3.6.3

    Topologists know that "there are no knots in 4D." That is, all 1D loops are equivalent by distortion to the unknot in 4D space. Here is a mathematical knot; you should imagine it as 1D or infinitely thin. Show that you can unknot it by moving part of it into 4D. Like always, you can choose a series of drawings or you can try to describe it using sentences. Video hint at the bottom.
    diagram of trefoil knot

    Question 3.6.4

    Here is a different kind of image of the tesseract. It is all of the edges of the tesseract, and it shows some of the 2D faces, but it doesn't show any of the 0D, 3D, or 4D parts. It is accurate in a way, even if it is sort of bent out of shape so it fits in 3D world. I like this one because I feel like it's more obvious that there are 16 vertices and 24 edges in this picture.

    blue translucent immersed hypercube, all vertices and edges showing.

    Image credit Mouagip.

    Your task is to describe or draw the locations of three of the 8 3D cubes in the boundary of the tesseract. Remember, they will be really bent out of shape. That's okay.

    Question 3.6.5

    Let's look at our family of cubes and look at a new definition of distance. We will choose any two vertices on the object, and the distance between those two vertices is the smallest number of edges that it takes to get from one vertex to the other if you're only allowed to travel on edges.

    For example, on a 2D square, some pairs of vertices are adjacent, so the distance between them is 1. Two pairs of vertices are opposite each other, so their distance from each other is 2. There are no pairs of vertices that have distance higher than 2. So 2 is the biggest distance between any two pairs.

    • In a 1D cube, what is the biggest distance between any two vertices?
    • In a 3D cube, what is the biggest distance between any two vertices?
    • In a 4D cube, what is the biggest distance between any two vertices? (Hint: you can use the picture from #11 to help prove your claim).
    • In an n-dimensional cube, what is the biggest distance between any two vertices? (This one is just a conjecture, you don't have to prove it).

    Helpful videos

    Check out Beyond the Third Dimension for some excellent help on the torus problems.