3.5 4D Stacking and Slicing
|TCCNS Course||MATH 1332: Contemporary Mathematics|
|UT Austin Course||M 302: Introduction to Mathematics|
Suggested Resources and Preparation
Materials and Technology
Manipulatives can be helpful such as
objects that can be sliced, such as apples, tomatoes, floral foam, bread
one of those paper slicers for teachers, or a rotary cutter and straightedge, or just an Xacto blade for slicing paper objects
Completion of 3.4 4D Basics and Skills
Overview and Student Objectives
Students will understand that:
- Level slices can help us understand objects more deeply, by allowing us to see the interior of 3D objects or visualize the structure of 4D objects.
Students will be able to:
- Draw multiple level slices of 2D and 3D objects in the plane (a level slice being the intersection of the object with a horizontal plane).
- Draw multiple level slices of 1D objects embedded in the plane on a number line (a level slice being the intersection of the object with a horizontal line.
- Look at a set of level slices and predict what the object could be.
4D: Stacking and Slicing
Building objects by stacking, dissecting objects by slicing
We will try to understand n dimensional objects by thinking of them as stacks of infinitely many objects, most of which are exactly one dimension lower. So the questions I ask you will look like:
- If I show you five slices of an object out of the stack, can you guess what the object would be?
- If I show you an object, can you imagine slicing it and draw several slices?
If I take a 2D sphere and make a pair of horizontal slices, I get a ring. If those horizontal slices are infinitely close together, my little slice is an infinitely thin ring, AKA a circle. We usually say "the intersection of a sphere with a horizontal plane will be a circle."
As you move from top to bottom, you'll get circles of different sizes. The topmost circles are tiny, they grow until they get to the biggest circle, the equator, and then they shrink again until you get to the South Pole.
On the other hand, suppose you were taking very thin slices of a mathematically thin plastic straw.
Each little slice is still a single circle. The difference is that these circles will all be the same size. So looking at multiple slices gives us extra information about what the object could be.
The Triangle Family
In math, we have "families" of objects. Sorting things into families helps us understand higher-dimensional objects that we can't see. For this class, I want you to learn about two families: the family of triangles (or simplices) and the family of cubes. I will present triangles and cubes to you in a couple different ways, since different people like different explanations.
|1||0, 1, 2, 3|
|2||1, 2, 3, 4|
If you take level slices of a 1D object, then most level slices should have ___ dimension(s).
If you take level slices of a 2D object, then most level slices should have ___ dimension(s).
If you take level slices of a 3D object, then most level slices should have ___ dimension(s).
If you take level slices of a 4D object, then most level slices should have ___ dimension(s).
If you are looking at an object's level slices and most of them are 0D, then you know the object will have ____ dimension(s).
If you are looking at an object's level slices and most of them are 1D, then you know the object will have ____ dimension(s).
If you are looking at an object's level slices and most of them are 2D, then you know the object will have ____ dimension(s).
If you are looking at an object's level slices and most of them are 3D, then you know the object will have ____ dimension(s).
Here are five 1D level slices of a 2D object. What could that object look like? Select all that apply.
These exercises always have infinitely many correct answers. Object is a circle or diamond or oval or something like that. Object 2 is a triangle. Object 3 is a figure 8, or perhaps two diamonds that are stacked on top of one another and touching. Object 4 is a zig-zagging line segment or an S curve. Object 5 is a hollow square or rectangle shape.
Object 1: sphere, christmas ornament shape, or similar. Object 2: hollow box or cube. Object 3: hollow tetrahedron. Object 4: torus. Object 5: Hourglass shape, hollow.
Object 1: solid 3D ball or solid christmas ornament shape. Object 2: square-based pyramid. Object 3: rectangular solid. Object 4: vase or water glass. Object 5: Octahedron or two pyramids squished together.
A 3-D sphere can be thought of as a stack of infinitely many 2-D spheres. As you move from top to bottom in the fourth dimension, the 2-D spheres grow to the center or "equator" sphere and then shrink again.
Top slices should be a single circle. I hope they ignore the belt loops and pockets. There is one slice that is a figure 8. All lower slices are pairs of circles.
A single slice on top and bottom is a single circle. All other slices are pairs of concentric circles. They don't have to draw the topmost and bottommost slice. The slices should have some accuracy relative to each other as in they spread out and then come together again.
Topmost and bottommost slices are single dot. Exactly two slices are figure 8s. Between the dots and figure 8s you get single circles. Between the two figure 8s you get pairs of circles next to each other. Ideally, they're picking a variety of slices from different parts of the torus but that's not specifically required.
Reflective writing. Responses vary.
Here are level slices of different 1D objects in 2D space. This means we took a 1D object and sliced it at different heights with a horizontal line. What could these objects be?
Here are level slices of different 2D objects in 3D space. This means we took a 2D object and sliced it at different heights with a horizontal plane. What could these objects be?
Here are level slices of different 3D objects in 3D space. This means we took a 3D object and sliced it at different heights with a horizontal plane. What could these objects be?
Family of spheres! A 2-D sphere can be thought of as a stack of infinitely many circles. As you move from top to bottom, the circles grow to the equator and then shrink again. Here's an analogy exercise for you:
- A 0-D "sphere" is two dots. (Not one, but two).
- A 1-D "sphere" is actually a circle. It can be thought of as a stack of infinitely many 0-D spheres. As you move from top to bottom in the second dimension, the 0-D spheres grow and then shrink again.
- A 2-D sphere can be thought of as a stack of infinitely many circles. As you move from top to bottom in the third dimension, the 1-D spheres grow to the equator and then shrink again.
Using this pattern, describe a 3-D sphere (like, just write those two sentences down but increase every dimension by 1). Then briefly describe your reaction to what you wrote. Is it impactful or meaningful? Or just confusing? Why?
Draw what you get if you slice this pair of pants at 7 different heights. Imagine that the pair of pants is infinitely thin; this means your drawings should be of 1-D objects. You can do a top-down perspective if you want, or you can do an angled perspective like the 2-D sphere in Question 4.
Let's rest a torus (a hollow 2D donut) on a tabletop and then slice it with horizontal slices. Draw at least five of those 1D slices.
Now let's hold a torus (infinitely thin hollow donut) vertically as shown and slice it horizontally. Draw at least five of those slices.
Question 3.5.8 - Reflective Writing
Set a timer for five minutes. For that full five minutes, think about and write about what kind of person you want to be in the future. How can/will studying in school help you achieve these goals for yourself?