## Axonometry: Applying Complex Numbers to Art

- Lesson

In this lesson, students examine and draw representations of cubes and then learn how to analyze these representations using complex numbers. Students use what they know about operations on complex numbers to see if a drawing is an accurate representation of a cube. They also learn how to generate complex numbers that will produce such representations.

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. 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. 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. 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.

- Cubes having an edge length of at least 2 inches (1 per student)
- Dot paper
- Protractor
- Compass
__Axonometry Overhead____Axonometry Activity Sheet 1____Axonometry Answer Key 1____Axonometry Activity Sheet__2- Axonometry Answer Key 2

**Assessment Options**

- Use Gauss' theorem to see if the points
*A*(3, 6),*B*(2, –3) and*C*(6, –2) generate a cube. Then look for a pattern in the coordinates of these points. Use the pattern to generate other numbers that also the pattern always work? - Give students the following 3 points:
*A*(–55, 148*i*),*B*(51, 94*i*), and*C*(160, 20*i*). Have students create a graph of their "cube" based on these three points. Does the picture seem like an accurate representation? Now have them calculate*a*^{2}+*b*^{2}+*c*^{2}. Is the answer "close" to zero? Discuss what "close to" mean in terms of complex numbers.

**Extensions**

- Have students find an example of a cube in a magazine or
printed from a Web page. Ask them to attach it to a complex grid on graph paper
and write down the three generating points,
*a*,*b*, and*c*. (They may need to enlarge the image with a photocopier to have better accuracy.) Have them use the*a*^{2}+*b*^{2 }+*c*^{2}= 0 formula to check the reasonableness of the representation. - Have students research the use of projections in technology, especially in computer animation. Have students discuss how 3-dimensional objects are represented 2-dimensionally, and how mathematics is involved in computer animation.
- Have student explore where
*a*^{2}+*b*^{2}+*c*^{2}is less than or greater than zero. Have students create three points where*a*^{2}+*b*^{2}+*c*^{2}> 0 and three points where*a*^{2}+*b*^{2}+*c*^{2}< 0. Have them explain what these values mean in terms of a rectangular prism projected upon a page. [It means that the object being projected is not a good representation of a cube.]

**Questions for
Students**

1.What happens to a vector in the complex plane when it coordinates are multiplied by *i*?

[A new vector of the same magnitude is created and it is perpendicular or orthogonal to the original vector.]

2. What is the difference between drawing a box (or a rectangular prism) and drawing an actual cube?

[A cube is a type of rectangular prism with congruent edges and faces while other rectangular prisms don't have to meet these criteria. Any axonometry drawing on paper that is supposed to be an actual cube should somehow show that the edges and faces are congruent.]

3. What are the similarities and
differences between squaring a complex number such as 2 + 3*i* and a binomial?

[The binomial 3

x+ 2 can be written 2 + 3x. When you square the binomial, you get 9x^{2}+ 12x+ 4. When you square the complex number you get 9i^{2}+ 12i+ 4 = 9(–1) + 12i+ 4 = –9 + 12i+ 4 = –5 + 12i.They may look different, but a complex number is a binomial, so squaring a complex number is the same thing as squaring a binomial.]

4. Ask students to state some patterns they noticed in the course of the activities.

[Answers will vary but students should note that when

Cis located on the imaginary axis,AandBare symmetric about the imaginary axis.]

**Teacher Reflection**

- How did students demonstrate that they understood the big picture of applying operations on complex numbers to understanding a particular geometric problem?
- In what ways did students seem motivated by this activity to want to explore and ask questions about complex numbers, art, or both? If students did not seem motivated, what could you have done differently?

### Learning Objectives

Students will:

- Draw representations of cubes using 3 generating line segments.
- Learn about spatial geometry by drawing different views of a cube.
- Plot complex numbers in the complex plane.
- Multiply complex numbers.

### NCTM Standards and Expectations

- Use Cartesian coordinates and other coordinate systems, such as navigational, polar, or spherical systems, to analyze geometric situations.

### Common Core State Standards – Practice

- CCSS.Math.Practice.MP1

Make sense of problems and persevere in solving them.

- CCSS.Math.Practice.MP4

Model with mathematics.