Author:
Kenneth DeMason
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

    3.7 Visualizing the Fourth Dimension

    3.7 Visualizing the Fourth Dimension

    Overview

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

    Introduction

    Overview and Student Objectives

    Lesson Length: 40-45 minutes

    Overview: This lesson details one way to visualize four dimensions using colored three-dimensional objects.

    Lesson Objectives:

    Students will understand that:

    • Color maps can be used to represent higher dimensional objects in one fewer dimension.

    Students will be able to:

    • Translate between color maps and their real-world representations. 

    Suggested Resources and Preparation

    Materials and Technology:

    • MATLAB can be used to supplement lecture. Code to generate the torus and Klein bottle, with color maps, is provided. 

    Prerequisite Assumptions: Lesson 3.4: 4D Basics and Skills and Lesson 3.5: 4D Stacking and Slicing.

    Suggested Instructional Plan

    Frame the Activity (5 minutes)

    • Ask students to reiterate the various ways we've studied 4D objects before -- by slicing, by viewing the fourth dimension as time, etc. 
    • One other way to understand four dimensions is to represent such objects with three dimensions and color, where color represents a specific slice in the fourth dimension.
    • Transition to the lecture, start by going down a dimension.

    Activity Flow (25 minutes)

    • Instructor is advised to follow the body as presented in subsequent sections. Italicized questions provide opportunities for student engagement.
    • Supplemental MATLAB code may be used to provide further examples and enhance clarity via an interactive demonstration. 

    Wrap-up/Transition (10-15 minutes)

    • Discuss quiz questions with students.
    • Introduce homework and work through Questions 1 and 3. For the latter provide a detailed example on how to describe an object in four dimension so students can mimic this explanation in the remaining questions.

    Overview

    This lesson details one way to visualize four dimensions using colored three-dimensional objects.

    Lesson Objectives

    You will understand that:

    • Color maps can be used to represent higher dimensional objects in one fewer dimension.

    You will be able to:

    • Translate between color maps and their real-world representations. 

    Visualizing the Fourth Dimension with Color

    The Klein bottle (depicted below) is an example of a 2D surface which cannot be embedded in 3D space -- it has to have a self-intersection.

     Klein bottle immersed in 3D with axes nearby to give some perspective. Klein bottle is colored with a gradient that ranges from orange to yellow to green to blue. At the circle where the bottle intersects itself, one part is yellow and the other part is dark blue.
    A 3D representation of the Klein bottle. Notice how the handle intersects the body near the top.

    Yet, we can actually view it as a surface in 4D space without any self intersections! But how can we possibly visualize something inside 4D space?

    To start, let's think about how to represent 3D data in 2D. As an example, we can consider contour maps, one of which is given below.

     2D plot with the third dimension depicted using color. Purple is used for lowest height, then the scale moves chromatically until orange represents the highest height. Left, a series of concentric rings. The outer ring is light blue and the smallest inner ring is purple. Right a series of concentric rings. The largest outer ring is green and the smallest inner ring is orange.
    A contour map depicting a peak and a valley.

    Here we have one valley (to the left) and one peak (to the right). The colors indicate the height above sea level -- more yellow indicates greater height above sea level whereas more blue indicates further below sea level. The spacing between the lines tells us how steep or shallow the peak/valley is -- the closer they are, the steeper.

    Question: Do you notice any limitations?

    Here's one: If you imagine yourself at any one of the colored points, you'll know your exact height relative to sea level. If instead you image yourself standing on any of the white spaces, you can only guess what it'll be. The solution is to add more contour lines, which fills in more and more of the white space. If you do this enough, you've colored in the entire map. An example is given below. We refer to these as color maps

      2D plot with the third dimension depicted using color. Purple is used for lowest height, then the scale moves chromatically until orange represents the highest height. The plot is mostly green (which is height zero) and a valley is depicted via shading through blue to purple. A peak is depicted using shading from orange to yellow.
    A color map with one peak and one valley.

    Not only is it much clearer that we have one peak and one valley, and what their exact shapes are, but we also have a way of telling height at every point on the 2D space above. We can represent 3D data as a coordinate (\(x, y, \) color), where the color signifies what height we're at according to the legend to the right. What we've effectively done is create a one-to-one correspondence between colors and real numbers. Below is the actual 3D graph of this data, which has been similarly colored. Notice how the color matches the height! This, however, is not a color map because it is the actual object.

     Same color map as above, but from an angle. The orange and yellow parts clearly form a peak, and the blue and purple parts form a valley.
    The 3D representation of the above color map.

    With this in mind, we can talk about visualizing 4D data in 3D space using color. If you have any surface in 3D space, you can similarly color it. Now, each point has the form (\(x, y, z, \) color), and you've turned it into a surface in 4D space! We can do this, for example, on the torus. 

     A torus with the same color gradient on it. This time the colors represent height in the fourth dimension.
    A color map for the torus.

    Now, the coloring no longer corresponds to the height in 3D because we are looking at a color map, not the actual torus. If you look the yellow spiral, it winds around the central hole, and passes both on top and below the torus! Even though the height along this spiral is changing, the color remains the same.

    Question: What does the color represent in the above color map?

    It's the height now in 4D space! Our torus is really sitting in 4D space somehow, and the coloring is telling us how it sits. Each color represents a different slice of 4D space.

    One consequence of this is if we change the coloring, we do not change how the torus sits in 3D space. But we do change how it sits in 4D space, it'll be rotated, translated, or otherwise distorted.

    Terms

    • A color map is a low dimension colored representation of a higher dimensional object, where the color depicts the extra dimension. 

    Quiz Questions

    Solutions

    Question 1

    Color Map3D Representation
    1C
    2A
    3B

    Question 2

    A hemisphere or dome.  

    Question 1

    The figures in the left column show color maps of 2D circles sitting inside 3D space. The figures on the right are the actual 3D representations of these circles. Match the correct figures on the left to those on the right.

    2D plot of circle with color gradient at right ranging from purple at bottom to yellow at top. Top and bottom of circle are purple, left and right ends of circle are yellow.3D angled perspective on a circle. From this perspective it looks like an oval.

    2D plot of circle with color gradient at right ranging from purple at bottom to yellow at top. Top and bottom of circle are green (middle height), left end is purple (lowest height) and right end is yellow (highest height).3D angled perspective on a curve that appears to peak three times and has three valleys as well.

    2D plot of circle with color gradient at right ranging from purple at bottom to yellow at top. Upper left and middle right and lower left points are yellow (highest). Upper right, middle left, and lower right points are purple (lowest).3D angled perspective on a curve that appears to have two peaks and two valleys, like a taco shell.

    Question 2

    Below is a color map of a 2D disk sitting inside 3D space. What 3D shape does it form?

     color map of disk with outer edge being purple (lowest height) graduating to the yellow (highest height) in the center.

    Homework Questions

    Solutions

    Question 1

    Although the \(x\), \(y\), and \(z\) coordinates of the intersecting points are equal, their colors are different (the handle is yellow but the jug is blue). Since color is representing height in the fourth dimension, these points are actually distinct.

    Question 2

    It may still have no self-intersection in 4D space, but it depends how it is colored. For example, if the Klein bottle is colored with only one color, then really we get the exact same representation in 4D space, since everything is at the same 4D height. So, the self-intersection will still be present. As long as the intersecting points are colored differently there will be no self-intersection.

    Question 3

    Part A

    The plane lies in 4D space. Even though it is a 2D object and can sit perfectly in 3D space, the coloring does not correspond to height in 3D space. The plane is tilted but a single color -- if color matched height in 3D space, we would see a gradient.

    Part B

    The plane is contained within an entire slice in 4D, and is tilted within that slice. By the coloring we know the slice corresponds to height 0 in the fourth dimension.

    Question 4

    Since this plane is also a single shade of color, it is at a single height in 4D space. It is still tilted in the same way, just at a different height -- now height 1. All we did is translate it up in the fourth dimension.

    Question 5

    Not only is the plane tilted in 3D space, but it is tilted in the fourth dimension as well. Imagine the fourth dimension as a vertical line, where each point represents a single 3D space. This is the 4D height. In Questions 3 and 4, our tilted plane occupied one of these single 3D spaces. Now, cross-sections of this plane have been dispersed throughout these 3D spaces. 

     

    Below is a depiction of the Klein bottle sitting inside 4D space. To achieve this, we have placed it in 3D space and colored it using a color map. The next two questions refer to this figure.

     Klein bottle immersed in 3D with axes nearby to give some perspective. Klein bottle is colored with a gradient that ranges from orange to yellow to green to blue. At the circle where the bottle intersects itself, one part is yellow and the other part is dark blue.

    A color map of the Klein bottle.

    Question 1

    Even though our 3D visualization of the Klein bottle has a self intersection (where the handle meets the top), the Klein bottle does not have any self-intersection in 4D space. Describe why.

    Question 2

    If the Klein bottle above were colored differently, would it still have no self-intersection in 4D space? Explain why or why not.

    Question 3

    The following figure depicts a color map of a plane.

     3D angled perspective on a plane. Entire plane is colored green (height 0).

    Part A

    In what space does this plane lie in?

    Part B

    How this plane geometrically lie in the space you answered in Part A?

    Question 4

    The following figure depicts a new color map. Geometrically, how does this new plane lie relative to the one in Question 3? 

      3D angled perspective on a plane. Entire plane is colored yellow (max height).

    Question 5

    Here is one more color map. Describe how this plane sits in the space you answered in Question 3 Part A. 

      3D angled perspective on a plane. Left end of plane is colored purple (lowest height) and is also the lowest part of the plane in the three visible dimensions. The right side of the plane which is also the highest part in the three visible dimensions is colored yellow (highest).