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, 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:
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.
In a 3 x 3 subdivided square, there are
__vertices
__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.
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 V
subdivisions
of each edge
1
4
2
9
3
4
E
S
V+E+S
V–E+S
4
12
1
4
9
25
1
1
What patterns do you see?
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.
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.
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.
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.
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 V
Subdivisions
1
8
2
27
3
E
S
C
12
54
6
36
1
8
Number of V + E + S + C
Subdivisions
1
27
2
125
3
V–E+S-C
1
1
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?
Number of V
Subdivisions
1
2
E
S
C
H
What about a hypercube subdivided so that each edge is
subdivided into three unit segments? What patterns can we
imagine in this case?