For this lesson, students must be familiar with graphing
complex numbers in the complex plane and operations with complex numbers.

In the first period, distribute a cube to each student. If you do not have cubes, then have students make their own origami cubes (instructions on how to do this can be found doing a simple search on the internet). Have
them observe the cube from different points of view. Next, have students
sketch, as best they can, their cube on a piece of dot paper from the
perspective of one vertex being closer to them than the others, and so that the
three edges (whose common endpoint is the vertex closest to them) appear to be the same length.

Then show the Axonometry Overhead.

__Axonometry Overhead__

Ask students,
"Without measuring, do the three edges that meet at the vertex closest to you appear to be the same length?" [Yes.] Have students hold their cube
in front of them, using the same perspective as in the overhead. Then ask them
to slightly tilt their cube, so that point *A*
moves away from them and points *B* and
*C* move closer to them. Ask,
"When you tilt the cube, does the appearance of the edge lengths
change?" The answer should be yes. If students aren't seeing that one of
the edges appears to decrease in length and two of the edges appear to increase
in length, have them increase the tilt of the cubes until they can see it. This
is a central idea of the lesson. Then ask, "Do the angles between the same
three edges appear to change as you tilt the cube?" [Yes.] This is a
secondary idea of the lesson. Ask, "Why don't the lengths of the edges
actually change?" [The lengths of the edges do not change because the physical shape of the cube is not changing. The edge lengths are constant.] Now say, "We are going to
mathematically explore the relationship of the three edges that meet at the
vertex closest to you."

Explain to students that the drawings of cubes
we are dealing with in this activity are the result of a 3-dimensional cube
being projected straight down onto a 2-dimensional plane at a 90-degree angle.
This is the same type of projection that happens with an overhead projector and
it is called an orthogonal projection.

The
mathematician Gauss realized that it is very useful to project the
3-dimensional figure onto a complex plane. A complex plane has one real axis
and one imaginary axis. A point on the complex plane is defined by its
coordinates, which represent a complex number. For example, the coordinate (2,
3) represents the complex number 2 + 3*i*.
In three dimensions, a cube can be fully specified by the coordinates of one
vertex and the three nearest vertices. When these three vertices are projected
onto a complex plane, the corresponding vertices in the projection can be
represented by the complex numbers *A*, *B*, and *C*. Gauss’s Fundamental Theorem of Axonometry states that if *A*^{2}
+ *B*^{2} + *C*^{2} = 0, then the drawing on
your paper is an orthogonal projection of a 3-dimensional cube above the paper.
In other words, it is a good representation of a cube.

To understand
why Gauss’ theorem is true, you can show that perpendicular vectors of the same
length can be generated in a complex plane by multiplication by *i. *So, for example, start with a vector
defined by point *A* at (3, 4*i*). The vector is defined with its tail
at the origin. Multiplying the coordinates of *A* by *i* gives 3*i* and 4*i*^{2} = –4. This defines a new vector with point *B *located at (–4, 3*i*). Students can try this with different
points and they will always find:

- The length (magnitude) of the vector for
*A* equals the length (magnitude) of the vector for *B.* The transformation does not change the
length (magnitude) because multiplying by *i *is
similar in some regards to multiplying the identity element by 1. - The vector for
*A* is always orthogonal to the vector for *B*.

Next, you can
show that for any two points generated this way, *A*^{2} + *B*^{2}
= 0. For the points above, *A*^{2}
= (3 + 4*i*)^{2}= 9 + 24*i
*- 16 and *B*^{2} = (–4 + 3*i*)^{2} = 16 – 24*i*
- 9. It is easy to show that these squares will always
generate three sets of opposites that sum to 0. In this case:

3^{2} and 3*i*^{ 2}

–4^{2} and 4*i*^{ 2}

24*i* and –24*i*

The general algebraic case goes as follows:

Assume *A* is located at (x, yi). Then *B*, found by multiplying *A*'s coordinates by *i*, is located at (-y, x*i*). *A*^{2}=x^{2}+2xy*i*-y^{2} and *B*^{2}=y^{2}-2xy*i*-x^{2}. From here, it is easily shown that *A*^{2} + *B*^{2}
= 0.

Next, apply
this analysis to the cube itself. Each face of the cube can be plotted in a
complex plane. For any two vertices that define orthogonal edges of the cube, *A*^{2} + *B*^{2} = 0. It is easy to show that for three vertices that
define the cube (all neighbors of a single vertex), *A*^{2} + *B*^{2}
+ *C*^{2} = 0 because *A*^{2} + ^{2} = 0 and *B*^{2}
+ *C*^{2} = 0.

Return now to
the orthogonal projection. In this lesson, students will observe that the coordinates
*a* and *b* in the orthogonal projection generally do **NOT** define orthogonal vectors. The coordinates in the projection are
orthogonal only when the complex plane chosen is parallel to one of the cube’s
faces. However, part of the genius in Gauss’s theorem is that we still benefit,
because properties of the 3-dimensional relationship still hold with the 2-dimensional
analog. The full proof is too long for this lesson, but in a nutshell, *A*^{2} + *B*^{2} + *C*^{2}
= 0 even when *A*^{2} + *B*^{2} ≠ 0 and *B*^{2} + *C*^{2} ≠ 0 in two dimensions.

A key point to
emphasize with students is that Gauss’s theorem holds only when the vectors of
the 3-dimensional figure meet two criteria: they have the same length (always true for a
cube) and they are orthogonal (always true for a cube). So, if the 3-dimensional
figure is not a cube, then *a*^{2}
+ *b*^{2} + *c*^{2} ≠ 0.

Explain that
students will evaluate the formula for the complex numbers that represent the
endpoints—*A*, *B*, and *C*—of the
generating line segments. If *a*^{2}
+ *b*^{2} + *c*^{2} = 0, then the 2-dimensional
drawing is an accurate representation of the 3-dimensional cube. Distribute the
Axonometry Activity Sheet 1.

__Axonometry Activity Sheet 1__

__Axonometry Answer Key 1__

Ask students to work
in pairs as they complete the activity sheet. Tell them to refer to the
overhead to help with the visualization described in the activity sheet.

Afterwards, hand out Axonometry Activity Sheet 2 to
each student.

__Axonometry Activity Sheet 2__

Axonometry Answer Key 2

Circulate as students work on the activity sheet. Remind them to
use their cubes to help them visualize the actions described in the activity sheet. The easiest way to find another set of points would be to extend the three edges of a cube made by the points in Question 1 by a constant factor (ex: double the lengths of all the edges).

When students have completed the activity sheet, have them
share their answers with the class. Guide the discussion around the
conservation of the total length of the three edges as the cube is tilted, and
the angle relationships between the edges as the cube is tilted. Ask, "Why
are the angle measures of the projection's edges greater than the angle
measures of the actual cube?" [The perspective of the projected cube
causes the angles of the projection to be greater in size than the angles of
the actual cube.] Students might also think that because the actual edge
lengths of a cube do not change, then the projection's edge lengths also should
not change. Explain that, according to how the cube is tilted, the projection
will cause some edges to shorten while others lengthen. However, the total
length of the projection's edges should not change.

End the class with a summary of the discussion. Ask
students, "Where in a real-world context might a 3-dimensional cube be
represented 2-dimensionally? " [Answers may vary from blueprints to
technical drawings.]

As the
final activity, have students suppose they will
create three points in which *a*^{2}
+ *b*^{2} + *c*^{2} is close to, but not exactly, zero. Have
students explain what this would mean in terms of the lengths of the line
segments. [They are all close to each other in length.] Ask them if one of the
points can be moved so that *a*^{2}
+ *b*^{2} + *c*^{2} = 0. Encourage students to conclude that the reason *a*^{2} + *b*^{2} + *c*^{2} = 0 for a cube is because cubes have equal
edge lengths.

**Reference**

Dörrie, Heinrich. 100
Great Problems of Elementary Mathematics: Their History and Solution.Dover
Publications; Softcover Edition, June 1, 1965.