Lesson Plan
Procedures
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This page covers specific steps to present a lesson on "How to Draw N-space
Cubes" and "Chasing Snakes in N-space Cubes". The files and programs are listed in the
Materials Required web page and are free for you to use in your classroom.
Dennis Clark, the author, retains all rights to the files and programs.
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Read the web page about Making Tesseracts: Pictures of an N-cube classroom experience.
It will give you a peek into an actual class where the subject was presented. You can
see materials that we used and how the students responded. Hopefully, it will enthuse
you to the point where you continue to pursue the project.
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Read the "How to Draw N-space Cubes" PowerPoint presentation. You can either view
the presentation online as 71 large, (200k each) slow-loading web pages or you can download
the entire presentation in one 439k PowerPoint file. However, you must have the Microsoft
Powerpoint program to view the downloaded PowerPoint file on your computer.
This presentation contains all of the information needed to understand how to draw
N-space cubes. However, there is a vanishingly small possibility that I wasn't as
clear as I thought in creating the presentations in which case I will happily answer
questions via email or telephone.
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Read the "Chasing Snakes in N-space Cubes" PowerPoint presentation. Again, you can
view the presentation online as 53 large (200k each) slow-loading web pages or you can
download the entire presentation in one 502k PowerPoint file. Of course, if you don't
have PowerPoint, you will need to view the pages online.
This second presentation contains all of the information needed to chase snakes in
N-space cubes. However, if you haven't read the first presentation, this one will
be pure gibberish. If you have read both presentations contact me to explain
any remaining gibberish.
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If you have actually reached this step you are to be commended. You are now an expert
cube-maker and snake-chaser ready to create a fascinating classroom experience
for your students. I guarantee they've never seen this stuff before! You must now
decide what parts of the presentation you wish to prepare for your class. Your available
resources will undoubtedly be the determining factor. The presentation can be made
with only the Group One materials. They are hopefully available to all teachers in
all schools.
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Tell the students that they are going to be able to draw 4 and 5-space cubes. Ask them
what a "cube" is. Collect all of the "facts" about a cube.
It's a three-dimensional, six-sided solid.
Each side is a square.
All angles are 90 degrees.
Each edge is of equal length.
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Ask for a volunteer to draw some pictures on the blackboard. (Seated students are
encouraged to make suggestions to help because "this is going to be difficult!"
"Now," pause significantly, "draw a --- POINT!"
Student probably draws a small, filled in circle on the blackboard.
Applaud the student wildly when he or she (referred to from now on as "she" or "they")
finishes the arduous task. Encourage the class to join in the applause.
"Next," pause "draw a --- LINE!" ... Applaud.
"Draw a SQUARE!" Applaud.
"Draw a CUBE!"
At this point the student may have a little difficulty. They may draw a squashed diamond
with three descending lines that attach to the corners of a "V". This is what they would
see in a picture of a square block of wood. Applaud. Then ask if they can show a
cube where all of the edges and corners are visible. (They may draw this second type of
cube to start with.) The second type of cube is usually portrayed as two overlapping diamonds
with corresponding corners of one diamond attached to the corresponding corners of
the other diamond. (two diamonds and four connecting lines) Applaud.
Announce that the volunteer has now created the first four N-space cubes (hypercubes)
where N = 0, 1, 2, and 3. They created a zero-space cube, a one-space cube, a two-space
cube and a three-space cube.
Point to each N-cube as you identify them.
"Now. Draw a --- TESSERACT!" Stupefied silence and "What do I do now" expressions.
After asking the audience for help - and none forthcoming - relent and backtrack and
announce that maybe the progression isn't quite clear yet. (It helps to be a ham about
all of this stuff.)
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Explain that within two minutes the students will be able to create a Tesseract (a 4-space
cube) without the teacher's help. "You'll even be able to make a FIVE-space cube!
And you'll be able to do it by yourself - without my help!"
Of course, this depends on how well you explain things in the next two minutes.
Be gutsy. Have them time you.
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Explain that a 0-space cube is a dimensionless point which you represent on the blackboard
as a little dot. As you make the following explanation, be certain to illustrate it
on the blackboard with appropriate figures. You will recognize this section from the
"How to draw..." presentation.
"Take two zero-space cubes and connect their corresponding points (or
corners) and you have a one-space cube. It's what we usually call a line. Take two
one-space cubes and connect their corresponding points (or corners) and you have a
two-space cube (a square). Take two two-space cubes and connect their
corresponding corners and you have a 3-space cube. (a cube). I'm finished!"
This is the end of your two minute time limit. Did you explain it in less than one minute?
The key is in using the same words each time you create the next higher n-cube.
"Now, " pause significantly "How do you create a 4-space cube?" Ask the whole class to
help the volunteer. Ask for the "algorithm" you have just made plain. The answer is
"take two 3-space cubes and connect their corresponding corners and you have a 4-space
cube". Help them describe it if they need it. Help the volunteer connect the corresponding
corners of one 3-space cube to another 3-space cube. Watch carefully because
it is easy to connect the wrong corners together. When she has finished ask
for applause from the audience as you applaud the volunteer. Let the poor student sit down.
Now you can ask if they can describe how to create a 5-space cube. "Take two 4-space
cubes and connect their corresponding corners and you have a 5-space cube."
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At this point the students will look at the messy 4-space cube on the blackboard and cringe
at what they will have to do to create a 5-space cube. In fact, you can comment on how
ugly a process that is going to be. It would be a real mess of crisscrossing lines.
Tell them that it looks like you need a better method of drawing n-space cubes.
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"Let's go back to the 3-space cube. You told ME that a cube has edges or lines that are all
the same length. The 3-cube on the board doesn't have all lines of equal length. You told
me that all of the angles in a cube are 90 degrees. That 3-cube on the board doesn't have
all 90 degree angles." (measure them if they don't believe you) "So how can you call that
thing on the blackboard a cube? It violates many of the geometric properties that you
stated make up a cube!" Fish around for a while. Ask the students to discuss it and see
if you can lead the discussion around to the following conclusions.
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Any time you represent an N-dimensional object in less than N dimensions, the representation
will not accurately portray the object. Thus, a three dimensional cube displayed on a
two-dimensional blackboard will distort of some of the geometric
properties of that cube. Note that we "cheat" every time we draw a figure on the board.
Lines are not drawn with zero width. Points are not drawn with zero dimensions. What other
things do we distort? We have this fascinating mental ability to ignore and distort some
facts in order to make other facts clear. It's called Meta-thinking. We are so good at it
that occasionally we have to carefully think about it to keep from skipping over a lot of
tiny little details that in some contexts are important.
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We can't see a point that doesn't have any dimensions, so we cheat and make a little round
dot on the blackboard. We can't see a line with no width to it so we cheat and use a magic
marker that makes a line with a visible width to it. When we draw a "cube" on the blackboard
we can't really show three dimensions accurately, so we cheat and make a perspective drawing
that appears to recede into the two-dimensional blackboard. That makes the angles of the
cube appear to be other than 90 degrees. Some of the lines of the cube might not be the
same length as others because of the perspective drawing. When representing a 3-dimensional
object in two dimensions we have to distort some of the properties of the object. Likewise,
when we display N-dimensional objects in two dimensions, we have to distort more properties
of the object. This violates the rules of geometry. In the present case, topology rescues us.
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Topology relieves us of many of the geometric rules and properties of objects. Topology
is defined as a branch of mathematics concerned with those properties of geometric
configurations (as point sets) which are unaltered by elastic deformations
as a stretching or a twisting) that are homeomorphisms or the original figure.
Argh! I got that out of
the dictionary. But, what topology allows us to do is to make the physical arrangements
of points on an ncube unimportant. We can put the points of an ncube anywhere we
want as long as those points remain connected to same points they were originally
connected to. We can put the points of a cube in a circle and as long as those points
stay connected to the same other points - topology allows us to still call the resultant
figure a cube. That is, it is topologically equivalent. It looks vastly different, but
it is still a cube.
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So, we take advantage of the fact that we can put those "corners" of an ncube anywhere
we want - in a manner that suits our purposes. For our purposes, we want to arrange
the corners so that the resulting figure is pleasing to the eye. We want to arrange
them so that figure is as simple or clear as possible. Arranging the points sequentially
around a circle creates a figure that looks like (and is described as) a twisted
circle cube. When we look at the twisted circle cube we eventually discover that
if we untwist portions of the circle cube - we get something which is called (duh)
an untwisted circle cube. And, if we arrange the points using
the exclusive OR binary operation to define what points are adjacent to other points -
we get a figure where the number of points on a given horizontal level follow Pascal's
Triangle. Guess what we call That cube... a Pascal's Triangle Cube.
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You can see that the first figure has its points distributed around a circle.
In fact, the points are distributed sequentially around the circle. That forms
the twisted circle cube. The second figure's points were distributed around the
circle, but the points have been re-arranged so that the lines are untwisted.
The third figure's points are not distributed around the circumference of the
circle. They were created using the XOR binary operation. The red circles are
not part of the n-cubes. They are there just to show the circular arrangement
of the points in the first two ncubes. By the way, those are 3-space cubes.
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After the discussion of what an N-cube is, and getting the intuitive idea of how
to draw one, and after building an n-cube on the blackboard using the XOR operation,
you can end the lesson if you wish. You can assess what students have learned
by having them fill in the binary numbers on one of the hand-outs of
4 or 5-space cubes or you can challenge them even more by having them create a
5-space cube without a template. Extra credit can be given for anyone who wants to
create a 6 or 7-space cube by hand.
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The second "half" of the presentation introduces the students to a research
problem that, to my knowledge, has not been solved. The overarching question is:
How do you create a maximal length path (snake) through an n-cube of any dimension?
Ancillary questions are: How many unique snakes are there for any given n-cube?
What are the unique snakes for a given n-cube? How do we generate a maximal length
snake for any n-cube? These questions have not been answered by anyone. The students will
be able to understand what a snake is, and will be able to find 5-space snakes on
a 5-space cube handout.
A "snake" is a path that goes contiguously from point to point on any n-cube ending
up at the same point where it starts. Furthermore, (and this is the key point to
understand) no point on the path can be within one line length of any other point
that is on the snake except for a point's preceeding or succeeding point on the path.
A maximal length path is the longest snake that can be found in a given N-cube.
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The teacher can pretty much follow the steps outlined in the "Chasing Snakes in N-space
Cubes" PowerPoint presentation. When demonstrating what constitutes a valid maximal
length snake, the teacher can use a transparency on an overhead projector. (See the
templates section) An erasable magic marker can be used to outline a valid snake on the
transparency. Kleenex tissue can be used to erase "illegal" choices of points as
the snake is built segment by segment. Slightly wetting the kleenex makes it easy
to erase the segments. Students can then create their own maximal length snakes
starting with a 4-space snake. When they have found a snake they can check each
other's snakes as a classroom activity. When they have found a few 4-space snakes
they can graduate to discovering a larger 5-space snake. Again, they can check each
other's snakes as they find them. Bragging points are awarded for the first person
to find a valid 5-space snake. That snake can be taped or pinned up. The particular
snake can be added to a list on the blackboard. The idea is to have each student
find a snake. Some of them may need help from other students. That's fine as long
as they gain an understanding of what a valid snake is.
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When you wrap up the snake discovery activity you can go further into the PowerPoint
presentation and discuss ways of naming the snakes. The first method is to use
the binary names associated with each point on the snake. This is the first method
of naming snakes. It is a bit clumsy and it is easy to make mistakes with the binary
numbers. A second method of naming snakes is to use the "names" of the lines that
go from one point to another. Lines are named by figuring out which digit changes
when going from one point to another. If, in a 4-space cube, the second digit
changes from point 0000 to 0010 then the line between the two points is labeled
with a "2" and the first segment of the snake is named "2". If the subsequent
point on the snake is 1010 then the digit which has changed is ( 0010 to 1010 ) the
fourth digit and the snake is now named 2-4. A complete 8-length 4-space snake
might then be labeled: 0000, 0010, 1010, 1110, 1100, 1101, 0101, 0001 or
24321431.
x first digit
x second digit
x third digit
x fourth digit
4321
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0000 The binary labeled snake is a bit overwhelming.
2 0010 The line-numbered snake is both easier to view and remember.
4 1010 Additionally, patterns begin to appear as you find more snakes.
3 1110 For instance 23453142345314 was the first 5-space snake that
2 1100 I discovered by hand. I still remember it!
1 1101 Symmetries such as 12341234 begin to become apparent.
4 0101 Other interesting symmetries like 12342134 can be found.
3 0001 Can you find it in the snake 24321431 ?
1 0000
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A third way of naming snakes is given in the Chasing Snakes presentation
but it is a method best saved for hard-core puzzle fanatics.
In the meantime, you can discuss the two or three types of "translations"
that show which snakes are duplicates of each other. For instance,
12341234 is the same snake as 43214321 and 23412341. The Chasing Snakes
presentation discusses three types of translations. Students are encouraged
to see how many of the snakes that they have discovered are duplicates or
translations of each other. At the end of this activity you can show them
the list of unique snakes in 3,4,5, and 6-space. You can show them the
data file that contains over 10,000 7-space snakes. No one knows how many
7-space snakes there are, and no one knows how many unique 7-space snakes
there are. The data file with 10,000 snakes is not a list of unique snakes.
It will take a computer program to sift them down to an incomplete list of
unique 7-space snakes.
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As a conclusion to the session you can demonstrate why the computer program
that I wrote to find n-space snakes failed to complete its job. The Tailwag program
tries to go through all of the possibilities of creating a snake. For 3-space
the maximum length snake is 6. The program starts at point 000 and has three
points that it can choose next. (001, 010, and 100) It chooses one of those
points (001) and then has another three points that it can choose. (000, 011,
101) One has already been used, (000) so it chooses one of the other two points.
It continues in that fashion creating a bigger snake and sometimes having to
backtrack when it runs into parts of itself and eventually builds a big enough
snake to qualify as a maximal length snake that closes on the first point. It
writes that snake into a data file, then it backtracks a little and continues.
Eventually, it will finish with all of the possibilities. Well, perhaps.
N- max Number
space length of Tailwag
snake possibilities
3 6 3**6 729
4 8 4**8 65536
5 14 5**14 6103515625
6 26 6**26 170581728179578208256
7 48 7**48 3.67 e+40 (This is a Really Big number!)
8 ??? 8**?? unknown and astronomically large
Obviously, a brute force and ignorance program that tries to go through all of the
possibilities will fail because the task is just too large even for a computer program
working for hundreds of years. However, there are students in your class who will
very likely solve even more complicated problems than these in their lifetime. If
you can whet their appetite for solving the unknown, you're a wonderful, precious teacher.
Good Luck!