Name______
Date______
HEX EXPLORES THE SUBDIVIDED SQUARES
“Grandfather,” Hex asks, “What exactly is a dimension?”
When A Square begins to answer Hex, he starts with a single point, with zero dimensions, and moves it three units to trace out a segment subdivided into three unit segments.
The subdivided edge has four vertices and three unit segments.
He then moves the subdivided edge in a perpendicular direction, keeping it parallel to itself, [MD1]to trace out a square region, subdivided into unit squares. Hex figures out the number of squares, namely nine.
Here is a 3 x 3 subdivided square:
[MD2]
In addition to counting the number of squares, Hex can learn more about the subdivided square by figuring out the number of vertices and the number of unit segments.
[MD3]
In a 3 x 3 subdivided square, there are
__vertices[MD4]
__horizontal unit segments
__vertical unit segments
__total number of unit segments
__unit squares
To make things clearer, A Square can work out a specific case, for example the 2 x 2 subdivided square, with each two unit segments on each side.
[MD5]
The number of vertices is V = 9, three on each of the three horizontal rows. The number of unit segments is E = 12, with 6 vertical segments and 6 horizontal segments. The number of unit squares is S = 4.
Now draw a 4 x 4 subdivided square. How many vertices does it have, and how many unit segments and how many squares?
Compile your answers for the 3 x 3 case and the 4 x 4 case in the following table. The first and second rows are already filled in.
Vertices = V Unit Segments = E Unit Squares = S
Number of subdivisions of each edge / V / E / S / V + E + S / V – E + S1 / 4 / 4 / 1 / 9 / 1
2 / 9 / 12 / 4 / 25 / 1
3
4
[MD6]
What patterns do you see?
[MD7]
Can you predict what some of the numbers will be for a 5 x 5 subdivided square? Draw a diagram and use it to check your predictions.[MD8]
ON TO THREE-DIMENSIONAL SPACE
Hex isn’t satisfied just with finding patterns in the plane. She suggests to her grandfather that it is possible to go further and look for patterns in a space of three dimensions. What happens in our space of three dimensions? Moving a 3 x 3 subdivided square would produce a 3 x 3 x 3 subdivided cube. Even though Hex can’t see it, she uses her imagination to predict that it will have 27 unit cubes.
[MD9]
We in Spaceland can see a subdivided cube and if we separate the cubes slightly, we can see what is inside. We can see the 27 unit cubes, in 3 layers each with 9 unit cubes.
Hex found patterns in the plane by counting vertices and unit segments and unit squares in a subdivided square. In space, we can count vertices, unit segments, unit squares, and unit cubes in a subdivided cube.
[MD10]
As in the case of the plane, we are aided in our counting of the edges by considering each direction separately. As before, we get some good information by looking at a simpler case. For a 2 x 2 x 2 subdivided cube, we have 8 unit cubes.
[MD11]
Next, we have 27 vertices, 9 in each of 3 horizontal planes.
e
We have 54 unit segments, 12 in each of the three horizontal planes and 18 vertical unit segments, 2 for each of the 9 vertices in the horizontal plane.
Hardest to count are the unit squares. There are 3 horizontal planes, each with 4 squares, and 6 vertical planes, each with 4 squares. We have a total of 36 unit squares.
Once again, we can enter our numbers in a table and see what patterns we can recognize.
Vertices = V
Unit Segments = E
Unit Squares = S
Unit Cubes = C
Number of Subdivisions / V / E / S / C1 / 8 / 12 / 6 / 1
2 / 27 / 54 / 36 / 8
3
[MD12]
Number of Subdivisions / V + E + S + C / V – E + S - C1 / 27 / 1
2 / 125 / 1
3
THE FOUR-DIMENSIONAL CHALLENGE
At the end of Flatland the Movie, Hex challenges everyone to imagine what would happen in a fourth dimension.
In an earlier worksheet we have found that in a hypercube where each edge is one unit segment, there are V = 16 vertices, E = 32 edges, S = 24 unit squares, C = 8 unit cubes, and H = 1 unit hypercube. What do we get for a 2 x 2 x 2 x 2 subdivided hypercube, where each edge is subdivided into two unit segments?[MD13]
Number of Subdivisions / V / E / S / C / H1
2
What about a hypercube subdivided so that each edge is subdivided into three unit segments? What patterns can we imagine in this case?
[MD1] If color is available, we can ask students to count the number of red edges and compare it with the number of black edges. If worksheets only have grayscale, we can count the number of vertical edges and the number of horizontal edges.
[MD2]In the movie, when A Square constructs a diagram using square blocks, he separates them slightly. How will this help Hex to understand the geometry of the situation?
[MD3] Removing the squares makes it easier to count the number of edges.
[MD4]Students can count the number of vertices in different ways and they can discuss their methods. Determining edges sometimes is more confusing because students often lose count. Putting a tick mark on an edge as it is counted is a way that helps some students make accurate counts.
[MD5]The 2 x 2 case would be an easier place to start, but both the book "Flatland" and "Flatland the Movie" start with the 3 x 3 case.
[MD6]Pre-algebra students can discuss patterns in words. Students who have studied some algebra can express patterns as formulas in terms of n, the number of unit segments in an edge. Advanced students can prove their conjectures using mathematical induction.
[MD7]Many students, in middle school and above, will recognize that the numbers in the V and S columns are squares of integers and that the numbers in the V + E + S column are squares of odd integers. The V - E + S column will always be 1, an instance of the Euler characteristic of the subdivided square region, something that some students might have seen or might see in the future in other discussions.
[MD8]A good additional exercise at this time is to carry out the same investigation for subdivisions of an equilateral triangle. Although this works very well in the plane, it does not generalize easily to the third dimension.
[MD9]Separating the cubical blocks makes it easier for us to see what is happening inside the subdivided cube.
[MD10]The vertices are arranged in four horizontal planes, each of which is a 3 x 3 subdivided square. In addition to the edges in these horizontal planes, there are a number of vertical edges, each divided into three unit segments.
[MD11]Working out the 2 x 2 x 2 case in detail can help develop a strategy for the 3 x 3 x 3 case.
[MD12]More advanced students can fill in the 4 x 4 x 4 case and conjecture a general formula.
[MD13]Even the 2x2x2x2 case will be a difficult challenge for most students. The number of edges in a hypercube is discussed in an earlier worksheet and in one of the features on the Flatland the Movie DVD.