The square root of a reflection?

What things are squares? This question often leads to interesting and strange new worlds in mathematics.

We begin our story with whole numbers. If the whole universe you can consider is whole numbers, then the only squares are the “perfect squares” which we obtain by finding the area of a square with whole number sides.

Being unsatisfied with only some of our numbers being squares leads to a new kind of number: the irrational square roots of whole numbers which are not perfect squares.

So in the realm of real numbers, we have that to be a square it is necessary and sufficient to be non-negative. To have all real numbers be squares, we must again extend to complex numbers. For numbers, the story stops there. All complex numbers are squares. But is that the end of the story?

Certainly not. The question of squares in finite fields, for example, leads to the beautiful result of quadratic residues. But I want to consider squares under another operation: composition.

We can think about numbers as corresponding to the action of multiplying by that number. So 2 represents the function f(x)=2x. Then the square root of that action is another action, which we call \sqrt{2}, which when repeated twice, is the same action as 2.

Given a function f:A \rightarrow A, does there exist a function r: A \rightarrow A such that r \circ r = f?

In general, this functional equation could be difficult to solve, so let’s consider the case when A is the plane. Given a transformation F:\mathbb{R}^2 \rightarrow \mathbb{R}^2, can you find another transformation which is its “square root.” i.e. T(T(x,y))=F(x,y)

The first question, is what if F is a pure motion, such as a translation, rotation, reflection, or dilation.

If F is a pure translation by a vector v, this is easy: T should be translation by a vector \frac{v}{2}. Similarly, if F is a pure rotation, then simply by rotating by half the angle, you obtain T.

Dilation yields the original square root: the original F is a dilation with scale factor s, then T should be dilation at the same center, with scale factor \sqrt{s}. In fact, these three actions can all be expressed as operations on complex numbers. Addition yields translation; multiplication, dilation and rotation.

Here’s the interesting one: what’s the square root (with respect to composition) of a reflection?

One answer is that it doesn’t exist. In order to make the question well-posed, we must specify what kind of thing T must be. If T is required to be a similarity transformation, then it suffices to consider the determinant. By the equation det(T^2)=det(T)^2=det(F)=-1, it’s clear that no real value of the determinant of T will be possible.

But what if T is not necessarily a similarity transformation, but some other more exotic function of the plane? I don’t know the answer to this question, but my suspicion is that it is not possible for a reasonable function. I believe that the above argument extends to smooth maps via the total derivative.

Intuitively, though, we can imagine “rotating” the plane 90 degrees along the reflection axis, which repeated twice would give the original reflection. This of course, means T is no longer a function on \mathbb{R}^2. But by stretching the target, we can make a sensible choice for this “square root of a reflection”

If the original F was reflection in the y-axis, we could represent this rotation using a complex matrix: T=\begin{bmatrix} i & 0\\ 0 & 1 \end{bmatrix}. Then T^2 = \begin{bmatrix} -1 & 0\\ 0 & 1 \end{bmatrix}, which is indeed a reflection across the y-axis. T is now a function from \mathbb{C}^2 to itself. But since it leaves the second copy of the complex numbers fixed, we can visualize the action has happening in 3-space, where the first complex coordinate is represented by the xz-plane and the real part of second coordinate is represented by the y-axis.

Exploding Dots, Spy Codes and Minicomputers

When I was in fourth and fifth grade, my school used a math curriculum called “Comprehensive School Math Program” (CSMP). CSMP is one of the infamous “New Math” curricula developed in the 1960s. I expect I was one of the last few classes to use this curriculum.

I had occasion to revisit the CSMP materials after becoming fascinated by the phenomenon of Exploding Dots. In particular, I was struck by its similarity to some of the representations or “languages” of CSMP.

There are two points of convergence between Exploding Dots and CSMP: the abacus and the minicomputer. (It appears there is not a direct lineage according to statements by James Tanton on Twitter). The abacus is essentially isomorphic to Exploding Dots, while the minicomputer is related, but used much more thoroughly in CSMP.

The first is a representation in CSMP called the “abacus,” which comes in different forms (bases). For example, this is a task from the first semester of fifth grade:

CSMP Intermediate Grades III, Lesson N30

This task shows addition of fractions with unlike denominators. At the time of writing, while Exploding Dots contains a decimal experience, and there is a discussion of division on other machines, no equivalent to the above task seems to be included. The process of “trading” or “exploding/unexploding” is equivalent in the two systems. The abacus uses a grey bar instead of a point to separate the ones from the negative powered place values. This has the advantage of not needing the word “decimal,” for numbers in other bases, which is a bit of a contradiction. (Dozenal advocates use the semicolon as the separator to distinguish base 12 numerals)

CSMP introduces the ternary abacus with a very Cold War story about spies and sending secret codes.

CSMP Intermediate Grades III, Lesson N15

This script has an interesting interpretation of base 3 numbers: as functions from a finite set to a set of 3 elements. The “encoding” of the function is essentially the conversion process from base 3 to base 10, while “decoding” is the reverse.

The second representation that has similarities to Exploding Dots is the minicomputer. This is actually the representation that I recalled from my elementary days, as it was used consistently in the curriculum, whereas abacuses are more occasional.

The Papy Minicomputer is a chimera: a blend of base 2 and base 10. It is often introduced using the Cuisinaire rod colors. Checkers on each box are worth that value.

A single minicomputer is base 2. But after you continue, the values increase as base 10. The values of the second set of boxes is 10,20,40,80.

For example, the value of the above on the minicomputer is 800 + 20 + 4 + 1 = 825

The interesting thing about the minicomputer is that there are multiple correct ways of representing a single number, without using multiple checkers on a space. Similar to “antidots” in Exploding Dots, there exist negative checkers (notated with the “hat” symbol). Minicomputers also allow checkers with different values.

The archive of CSMP materials can be found at http://stern.buffalostate.edu/ Of particular interest is the videos of Frédérique Papy teaching children using the minicomputer and other languages of CSMP, found on this page: http://stern.buffalostate.edu/Movies/index.html

The Quadrilateral Zoo: Why Trapezoids Don’t Belong

I’m a resident teacher this year, and I’m working alongside an experienced teacher in a 10th grade Geometry class. During the unit where we discussed polygons and their properties, I came across this definition in the textbook:

A kite is a quadrilateral with two pairs of consecutive sides congruent and no opposite sides congruent

Pearson Geometry Common Core

This definition seemed…off. Why have that second condition?

I think the idea is to exclude the case where 3 sides are all congruent to each other and the other side is not.

Ceci n’est pas un cerf-volant

But why not just say “two disjoint sets of consecutive sides congruent” instead of “no opposite sides congruent”? The problem is that according to the Pearson definition, a rhombus is not a kite

Kite or no kite?

There is a famous debate in the definition of trapezoid, which is whether to use the exclusive or inclusive definition.

A trapezoid is a quadrilateral with at least one pair of opposite parallel sides

Inclusive definition

A trapezoid is a quadrilateral with exactly one pair of opposite parallel sides

Exclusive definition

The inclusive definition implies that a parallelogram is a trapezoid. In other words, the set of parallelograms is included within the set of trapezoids. And the inverse holds for the exclusive definition.

I decided to scientifically study this question, so turned to that time-honored rigorous methodology of…The Twitter Poll

No consensus apparently.

Many geometry books and educational resources have some kind of comprehensive picture of the classification of quadrilaterals. For example, the one from Wikipedia looks like this:

I do like this diagram, if only for the combination of a Venn diagram and actual examples of the quadrilaterals themselves. I notice two things: that the author is using inclusive definitions for kite and trapezoid, and the absence of the isosceles trapezoid.

Ready for it? Here’s my diagram:

What do you notice or wonder?

I arranged this diagram specifically to correspond to the subgroup lattice for the dihedral group of order 8.

Huh?

Basically, I’m classifying the quadrilaterals based on what portion of the full set of symmetries of a square that they exhibit.

Here’s the subgroup lattice diagram:

The notation used in this diagram is that “e” is the identity, “a” is a rotation by 90 degrees, and “x” is a reflection through two vertices, which fixes a pair of vertices and exchanges the other pair. There are two differences between this lattice and the one for quadrilaterals. First, there are two different subgroups of order two each corresponding to the kite and isosceles trapezoid. This is because the reflection symmetry that each exhibits can also be characterized as the other reflection that either goes through vertices or midpoints for the kite or isosceles trapezoid respectively. These two reflections are related by the rotation by 180 degrees, a.k.a. a^2. The other difference is that there is a cyclic subgroup of order 4 generated by a which doesn’t have a corresponding invariant set of quadrilaterals. This is because as soon as a quadrilateral has a 90 degree rotation symmetry, it is automatically a square. The reflections come for free!

So those teachers out there are probably asking themselves: where does this debate fit in the curriculum? Well, consider:

CCSS.MATH.CONTENT.HSG.CO.A.3
Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

Common Core State Standards for Mathematics

Ok, the standard doesn’t mention kites or rhombuses, which is a bit strange, but it still is clearly trying to get at a symmetry viewpoint of these shapes. In fact, the CCSSM has a thoroughly transformational perspective on geometry. The equivalence relations of congruence and similarity are consistently based in the transformation groups of rigid motions and dilations.

Interestingly, the standard says “trapezoid” even though a generic trapezoid has no symmetries at all.

In fact, what is up with trapezoids? As we saw, there’s some debate about the exact definition, but it always involves this parallel opposite sides idea. But we don’t need parallelism to define any of the other quadrilaterals. In fact, this points to a deeper fact: trapezoids don’t even have to exist! In spherical geometry, there are no parallel lines. So therefore, there are no trapezoids in spherical geometry.

Except, there are isosceles trapezoids. Because if we define an isosceles trapezoid as a quadrilateral with a reflection symmetry through the midpoints of opposite sides, then it exists perfectly well in spherical geometry. So here’s my hot take:

Isosceles Trapezoids are a more natural subset of quadrilaterals than Trapezoids.

When I say natural, I mean that it applies in a more general context, and fits more neatly in the symmetry classification. I consider the symmetry classification to be more consistent with the modern, transformational geometrical understanding than a classification based on sides and angles.

My friend Doug O’Roark pointed out that Zalman Usiskin has written an entire book on this subject, The Classification of Quadrilaterals : A Study of Definition. So this is not the end of this story. But for now, my current opinions are that trapezoids are weird, inclusive definitions are just better, and symmetry is a powerful and modern way to look at quadrilateral classification.

Aggregating Bi-variate Data in Desmos Activity Builder

I was creating an activity builder adaptation of a 3-Act plan called “Gas Station Ripoff” and I had a need to aggregate bivariate data across the whole class. [Original here. My version here.]

The only problem: aggregate only works on lists of numbers.

My work around was to add the following code:

code snippet for aggregating bivariate data in Activity Builder

What’s going on here? Well, “pump1point” is a mathematical input box. I get the latex content, and then parse this as an ordered pair object. Then, I get the x value (first coordinate). I then call the numericValue function so that aggregate can accept it.

Ultimately, what happens is I have a list called G_1 which contains the x-values for all the students. I do the same thing for the y-values, getting a list called P_1.

The advantage of doing it this way is that the CL eats the whole input at once, which means that student responses remain coupled, and in order when they are aggregated.

The final stage is to graph the list of points. After initializing each list, I simply put the following in the expression list of the graph:

The only drawback to this method is that students have to be precise about how they enter the data. It must be entered in the correct order. I’m not sure how robust parseOrderedPair is when there are missing or extra parentheses.

Please let me know if you find this useful, and comment with any questions. I am still learning the Computational Layer, so feedback from experts is appreciated.

Hello World

Here’s what you will find on this blog:

  1. A commitment to anti-racism and social justice.
  2. Reflections on being a new(ish) high school math teacher
  3. My thoughts about educational technology (GGB, Desmos, and TI most frequently)
  4. Free resources such as technology activities I create and lesson materials I have written
  5. Mathematics I am exploring as a learner (expository writing as an exercise to help me learn)