Lesson Plan

N-cube Materials

The materials below are organized roughly in order of their cost. Materials in the first group are the least expensive. All materials within a group may be necessary for that activity. Group One materials are necessary for the lesson. Materials in succeeding groups are optional.

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These items are used to create 2-D paper N-cubes, demonstrate XOR, and hunt N-space snakes.

three to five 8x11 sheets of paper per student. (two sheets used to xerox N-cubes)
one erasable magic marker per student. (Crayons don't work as well as markers)
one transparent 8x11 plastic paper protector per student.
one box of Kleenex tissue for the entire class for erasing magic marker lines
one blackboard or whiteboard with appropriate markers

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Materials in this group are handouts that can be used as classroom aids or assessment activities.

4-cube with no labels 4-space cube with lines and unnamed points
Students fill in appropriate binary numbers to the points on the handout.

5-cube with no labels "5-space cube with lines and unnamed points
Students fill in appropriate binary numbers to the points on the handout.

4-cube with no lines 4-space cube with binary point labels but no lines
Students draw the appropriate lines to create the 4-space cube.

5-cube with no lines 5-space cube with binary point labels but no lines
Students draw the appropriate lines to create the 4-space cube.

4-cube no lines or labels 4-space cube with unnamed points and no lines
Students insert binary numbers into the unnamed points (labels) and draw appropriate lines between adjacent points.

5-cube no lines or labels 5-space cube with unnamed points and no lines
Students insert binary numbers into the unnamed points (labels) and draw appropriate lines between adjacent points.

4-cube with lines and labels 4-space cube with labeled points and lines Can be used as a check to see how the students have done. Note that it is easy to create correct diagrams, lines and labels that are in different order than that shown here.

5-cube with lines and labels 5-space cube with labeled points and lines Can be used as a check to see how the students have done. Note that it is easy to create correct diagrams, lines and labels that are in different order than that shown here.

6-cube with decimal numbered labels 6-space cube with decimal labels
Used primarily to show that decimal numbers "hide" significant properties that show up when binary numbers are used. Decimal numbers do, however, fit into smaller labels. Hence, their use here.

TEXT FILE Unique Snakes data
Lists all of the UNIQUE snakes in 3,4,5, and 6-space cubes.

TEXT FILE 7-space snake data (10,000 of em)
Lists over 10,000 7-space snakes that have not been reduced to a subset of the unique 7-space snakes. The 10k snakes are only a small portion of the total found in 7-space.

Splash screen for Ncubes.exe Ncubes.exe Visual Basic program
A Visual Basic program that enables students to draw three types of 2-d N-space cubes. Twisted circle, Untwisted circle, and Pascal's Triangle arrangements of 3,4,5,6, and 7-space N-cubes.

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Materials in this group can be used to whet students' curiousity. They are used as advertising materials for the classroom experience.

8x11" bulletin board "ads" for the classroom session. For instance, "Chasing Snakes in N-space Cubes! (Coming soon to a math class near you)"
"Business card" advertisements could be circulated without comment or explanation. They might contain text like: "Snakes! 3 4 2 1 4 3 1 2 " The classroom session explains such cryptic notes.

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Materials in this group are used to construct one three-dimensional (3-D) tesseract. This tesseract will be a cube_within_a_cube type of tesseract.

20, thin, round dowels that are about two or three feet long. These dowels will serve as the "edges" of a 3-D tesseract. Several of the dowels are extra and will be used when some of them are accidentally broken. Can be found at a craft store.
Note: Straight sections of wire cut from coat hangers can be substituted for the dowels. Advantages: cheap, can be bent as needed. Disadvantages: Can cut students.
20 one inch diameter wooden balls. The balls will serve as the corners or vertices (also called points) of a 3-D tesseract. Four balls are extra. They will probably be used because it is easy to drill so many incorrectly placed holes in a ball that it becomes useless. They can be found at a craft store.
A drill with a drill bit that is as just barely as wide as the dowels. Used to drill holes at appropriate places in the wooden balls. Dowels will be stuck into the wooden balls and the result will be the 3-D tesseract. You might need an extra drill bit if the first is broken.

40, thin, round dowels that are about two or three feet long. These dowels will serve as the "edges" of a 3-D tesseract. Several of the dowels are extra and will be used when some of them are accidentally broken. They can be found at a craft store.
20 one inch diameter wooden balls. The balls will serve as the corners or vertices (also called points) of a 3-D tesseract. Four balls are extra. They will probably be used because it is easy to drill so many incorrectly placed holes in a ball that it become useless. Can be found at a craft store.
A drill with a drill bit that is as just barely as wide as the dowels. Used to drill holes at appropriate places in the wooden balls. Dowels will be stuck into the wooden balls and the result will be the 3-D tesseract. You might need an extra drill bit if the first is broken.

Note: Both types of Tesseracts (4-space cubes) can be created from other materials. For instance, straight sections of wire cut from coat hangers can be substituted for the dowels. Styrofoam balls can be substituted for wooden balls. They're cheap, and the n-cubes do not need to be drilled and pre-assembled before the class. However, n-cubes made from these materials are mechanically weak and tend to fall apart easily. Wires should be pre-cut with wire-cutters in order to avoid flying bits of trimmed wire. Pliers are useful to bend wires appropriately during the class.

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A wooden jig used to drill centered holes in the wooden balls These are materials to create a wooden jig that will be used to drill holes in all of the wooden balls mentioned in the previous groups. It is difficult to precisely drill holes in wooden balls. The jig helps keep the ball from moving and as holes are drilled, the ball is "pinned" with wooden dowels inserted through the jig and into the ball.

one 3"x3"x3" block of wood. The actual size of the block is not as important as that it be a square block of wood.
A drill press with a platform that can be tilted at exactly 45 degrees. The cube of wood must be positioned at a 45 degree angle on the surface of the drill press table in order to drill a hole in the corner of the cube at the right angle. This type of drill press can usually be found in a high school wood working shop. The jig could be created by a student as a special project.
A 1" hole is drilled down exactly 2" into one of the wooden cube faces.
A drill (the width of the dowels) is used to create three holes all the way through the center of each of the cube's sides.
The same drill bit is used to create a hole from one corner of the ball into the center of the ball. It will be easier to drill this hole if you sand the pointy corner down about 1/4" so that there is a flat surface for the drill to bite into. The photograph shows several large corner holes but only one small corner hole is needed.
The 1" wooden balls will be dropped into the 2" deep hole in the jig. The holes in the sides of the jig can then be used to precisely center holes approximately 1/4" deep into the wooden balls. After each hole is drilled into the wooden ball, dowels are inserted through the jig into the wooden ball. This keeps the wooden ball from accidentally moving as the next holes are drilled.
When all of the holes in the wooden balls have been drilled in the appropriate spots, the dowels are used to connect the balls together to create the tesseracts.

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A computer can be used to show the presentations and programs found on this web site.

A computer with a MS Windows operating system and the MS PowerPoint program.
A device to project a PowerPoint presentation on a large screen for the class.
The PowerPoint Presentation "How to Draw N-space Cubes" can be downloaded from this site. Free.
The PowerPoint Presentation "Chasing Snakes in N-space Cubes" can be downloaded from this site. Free.
The Visual Basic program: Ncubes.exe. It can be downloaded from this site. Free.

	4-space cube with lines and unnamed points Students fill in appropriate binary numbers to the points on the handout.
	"5-space cube with lines and unnamed points Students fill in appropriate binary numbers to the points on the handout.
	4-space cube with binary point labels but no lines Students draw the appropriate lines to create the 4-space cube.
	5-space cube with binary point labels but no lines Students draw the appropriate lines to create the 4-space cube.
	4-space cube with unnamed points and no lines Students insert binary numbers into the unnamed points (labels) and draw appropriate lines between adjacent points.
	5-space cube with unnamed points and no lines Students insert binary numbers into the unnamed points (labels) and draw appropriate lines between adjacent points.
	4-space cube with labeled points and lines Can be used as a check to see how the students have done. Note that it is easy to create correct diagrams, lines and labels that are in different order than that shown here.
	5-space cube with labeled points and lines Can be used as a check to see how the students have done. Note that it is easy to create correct diagrams, lines and labels that are in different order than that shown here.
	6-space cube with decimal labels Used primarily to show that decimal numbers "hide" significant properties that show up when binary numbers are used. Decimal numbers do, however, fit into smaller labels. Hence, their use here.
TEXT FILE	Unique Snakes data Lists all of the UNIQUE snakes in 3,4,5, and 6-space cubes.
TEXT FILE	7-space snake data (10,000 of em) Lists over 10,000 7-space snakes that have not been reduced to a subset of the unique 7-space snakes. The 10k snakes are only a small portion of the total found in 7-space.
	Ncubes.exe Visual Basic program A Visual Basic program that enables students to draw three types of 2-d N-space cubes. Twisted circle, Untwisted circle, and Pascal's Triangle arrangements of 3,4,5,6, and 7-space N-cubes.