A 5-space cube   Lesson Plan

    Overview

Students will learn how to use the binary math XOR (EOR) Exclusive OR operation to create N-Space Cubes otherwise known as N-cubes or hypercubes. Students will be able to demonstrate their ability to create 4-space cubes using a visual, geometric model enhanced by a mathematical tool. (The XOR operation)

Students are first shown an intuitive visual method of creating N-cubes. It is demonstrated on a black board using basic geometry. A definition of topology is introduced and students are basically shown how it differs from geometry. Once students have created four and five-space cubes with this intuitive approach they develop an appreciation as to why a better method of creating higher dimension N-cubes is desirable. (The N-cubes get too messy and complicated without a better, mathematical method.)

Students are then shown a mathematical way to create N-cubes. N-cubes have 2**n vertices (points or corners) where n = the dimension of the N-cube. So, a 3-cube contains 2**3 = 8 points. Also, each corner or point of the N-cube can be uniquely represented (named) by an n-digit binary number. Thus, a 4-space cube will have 2**4 = 16 points, each of which can be uniquely named using the sixteen, four-digit, binary numbers from 0000 to 1111.

At this point, students are given a definition for adjacent N-cube corners. Physically, adjacent corners are those that are one "edge" (one line) away from each other. Mathematically, adjacent corners can be said to be the corners that differ from each other by one and only one binary digit. Thus, the four-digit binary number 1101 is adjacent to the corners 1100, 1111, 1001, and 0101.

a 4-cube with binary labeled corners.

The mathematical operation to create adjacents (corners which are adjacent to each other) is the exclusive OR binary operation. To mathematically create the adjacents for a four-digit binary number, all one has to do is XOR the number with 0001, 0010, 0100, and 1000. The XOR binary operation is usually shown as: 

 XOR is defined as: 0 xor 0 = 0   Examples  1010    1010    1010    1010
                    0 xor 1 = 1             0001    0010    0100    1000
                    1 xor 0 = 1             ----    ----    ----    ----
                    1 xor 1 = 0             1011    1000    1110    0010

The exclusive OR operation is then repeatedly used to create the entire N-cube both mathematically, and visually. The method enables students to physically create larger (up to 7-space) N-cubes. As they draw them, students are shown how Pascal's Triangle is related to N-cubes. They are also shown that the co-efficients of the equation
(t+2)**N
gives the number of points, lines, squares, cubes, and sub N-cubes in any N-cube.

A comprehensive explanation can be found in the PowerPoint presentation "How to Draw N-space Cubes".
Students can then be shown a visual basic program that draws three different types of N-space cubes. (ncubes.exe) Other n-cube drawing programs can be found online.
The lesson can be concluded at this point if desired.

Chasing Snakes in N-space cubes

A detailed explanation can be found in the PowerPoint presentation "Chasing Snakes in N-space Cubes". The students are introduced to a research problem for which only a few answers are known. Students will find maximal length closed loops (paths) known as snakes in 4 and 5-space cubes. Most students are able to find at least one 5-space snake. They are shown maximal length snakes for 3 through 7-space cubes. They discover why it is relatively easy to find all of the maximal snakes in 3 through 6-space snakes and why the problem explodes computationally at 7-space and beyond. They learn how to recognize identical snakes even though they may appear to be different at first. They learn several different ways to name snakes and are shown the advantages of each method.

a 4-space snake
Even though students will discover many answers to this problem, they will discover that many mathematical problems remain unsolved.
The maximal length for an 8-space snake is not known.
There is no general formula or method for creating N-space snakes.
The total number of unique 7-space snakes is not known.
Unlike problems in a math book, the answers cannot be found in the back of the book. Even the teacher may not know the answers! As they explore the questions, they are entering unknown dimensions!