[Laser Cutting] Intro to Rhino and Grasshopper

I’m excited to share a brief, hands-on introduction to Rhino and Grasshopper–powerful software for 2D and 3D design work. Rhino is perfect for designing all sorts of objects–art, tools, toys, prototypes, etc. Rhino creates files that are ready-to-go on laser cutters and 3D printers. Rhino has a 90 day trial, and you’ll find that any skills you pick up in Rhino will transfer reasonably well to AutoCAD or other CAD softwares(Computer Assisted Design). Grasshopper is free.

Grasshopper is an easy-to-use add-on that allows you to easily create much more complex designs very easily–empowering users to make explicit use of mathematics within the design process. (More technically speaking, Grasshopper is a drag-and-drop functional scripting interface for Rhino.)

This intro is written to take you from 0 to On Your Way: if you follow along in the software, you should be able to precisely reproduce the pieces below, and you should understand the techniques well enough to get started making your own [insert your ideas here]!



What is an Algebra? (Part III)

In our next few Algebras, we broaden our sense of elements. Until now, we’ve looked at different types of numbers as our elements.

Now, consider matrices as elements. Given any two 2×2 matrices, A and B, we can certainly computer A+B and A\cdot B. And crucially, this sum and this product are still 2×2 matrices. So, we see that the set of 2×2 matrices forms an Algebra. It shouldn’t be too surprising that the set of 3×3 matrices also forms an Algebra, and more generally the set of nxn matrices do as well.

We could also use functions (say, ones that turn a real number into another real number) as our elements. In that case, we have addition and multiplication of functions as operations. And we have another reasonable operation to look into–composition.

Do we have division in the world of matrices or the world of functions? If you multiply a matrix A by a matrix B, is there another matrix to multiply by that will get you back to A? What about something that undoes composition by f or multiplication by f in the same way that subtracting f undoes adding f?



What is an Algebra? (Part II)

(This is Part II of a multi-part series. See Part I)

Consider the following statements about integer arithmetic.

even + even = even
even + odd = odd
odd + even = odd
odd + odd = even

even * even = even
even * odd = even
odd * even = even
odd * odd = odd

Speaking loosely, we can say that we have an Algebra consisting of the elements “even” and “odd.” We call this Algebra the Integers modulo 2. We write 0 as shorthand for even and 1 as shorthand for odd, and use the following notation to distinguish this modular arithmetic from our standard arithmetic.

0 + 0 = 0 (mod 2)
0 + 1 = 1 (mod 2)
1 + 0 = 1 (mod 2)
1 + 1 = 0 (mod 2)

0 * 0 =  0 (mod 2)
0 * 1 = 0 (mod 2)
1 * 0 = 0 (mod 2)
1 * 1 = 1 (mod 2)

A note on notation: think of the (mod 2) as describing the statement as a whole, not the element on the right-hand-side of the equals sign. What we are really saying is that in the Algebra of the Integers modulo 2, we have 1 + 1 = 0.


The following statements are a bit more of a mouthful, but are analogous to the example above.

[A Multiple of 3] + [A Multiple of 3] = [A Multiple of 3]
[A Multiple of 3] + [A number that is one more than a multiple of 3] = [A number that is one more than a multiple of 3]

Some simple notation is necessary. We use (mod 3rather than (mod 2) here to indicate that we are thinking about multiples of 3 rather than multiples of 2. We use to mean [A Multiple of 3], to mean [A number one more than a multiple of 3], and to mean [A number that is two more than a multiple of 3].

Then, we can write the statements above as
0 + 0 = 0 (mod 3)
0 + 1 = 1 (mod 3)

We leave it to you to write out the remaining 7 statements of addition and the 9 statements of multiplication. These statements describe the Algebra of the Integers Modulo 3.

So far, we have seen the Integers Modulo 2 and the Integers Modulo 3. It’s not hard to imagine the notation or interpretation for the Integers Modulo 4 or the Integers Modulo 37.

So if you are asked to find 2+2, you better know what Algebra you are working in! In the Algebra of the Reals or the Integers, 2+2=4. But in the Integers Modulo 3, we have 2+2=1(mod 3).

Given that our motivation is to understand what “An Algebra” is rather than to understand the intricacies of any particular Algebra, we’ll close our discussion of this family of Algebras here. The curious reader may want to explore whether division makes sense in this Algebra.

What is an Algebra? (Part I)

We’ll answer this question in this series of posts by giving a hands-on introduction to a variety of different Algebras (yes, there are lots!). Loosely speaking, an Algebra is a system that consists of elements and one or more operations. For now, you can think of the elements as numbers and the operations as the basic arithmetic you’re familiar with: addition or multiplication, for example.

So, our first example of an Algebra is the one we generally study in high school: the Real Numbers. And our operations are the basic four we know from elementary school.

Our second example of an Algebra is the set of integers. Here, our Algebra does not include all of our basic operations: division is problematic in that the quotient of two integers is not necessarily another integer. So, the Algebra of the integers includes addition, subtraction, and multiplication; but not division.

We’ll close this post with one more Algebra that is familiar from high school: the Algebra of the Complex Numbers. Addition, subtraction, and multiplication all behave nicely. Division is less clear to the novice eye: Given two complex numbers z=a+bi and y=c+di, does the notion of \frac{z}{y} make sense? Is it still a complex number? If the answers to these questions are yes, the Algebra of the Complex Numbers seems to be more like the Reals. And if not, the Algebra of the Complex Numbers seems to be more like the Integers. We’ll leave these questions about complex division for you to consider.

In the next post, we’ll consider an extremely useful family of Algebras: the Integers modulo n.


A Novel Take on Graphing

Here’s the pitch:

  1. Draw a horizontal axis and label it with the numbers 1,10,100,... (What numbers should go to the left of 1?)
  2. Draw a vertical axis, labelled similarly.
  3. Draw a downward sloping line at a 45 degree angle through the “origin” (which seems to now be the point (1,1). The equation y=-x no longer seems appropriate. Can you write an equation that fits this graph?
  4. Try going in the other direction. If you write down the equation that would normally give a circle and try to graph it on these axes, what do you get?
  5. Try lines, parabolas, circles, exponential functions, logarithmic functions, rational functions, etc.
  6. Try other ways of labelling your coordinate axes.

Note on Notation: My students and I reserve x and y for the standard axes labelled in the standard way. We generally use a to represent the strange values along the horizontal axis and b to represent the strange values along the vertical axis.


A Fractal Inducing Problem about Polygons

Here’s a question I like: take an n by n grid (start small), and draw straight lines connecting dots. If you end where you started and don’t cross lines, you get a polygon. What’s the most number of sides the polygon can have?

I received those words in an e-mail from Dan Finkel in April of 2014. I read the e-mail on the way from my front door to my car, and I visualized the 3×3 case for most of my commute to Tacoma.

Convinced that the problem would be perfect for my younger of two groups of elementary age homeschoolers that day, I launched into it–still with no clear ideas about the solutions. By the end of the hour, I still hadn’t had a chance to pick up a pencil, and I was skeptical that the this group of children had managed to produce the best solution–it’s hard to be satisfied by an asymmetrical solution to a problem like this. I pitched the problem to two groups that day and spent multiple hours after class exploring the different sizes–the 4×4, 8×8, 16×16 are particularly interesting to me for a particularly elegant solution. This simple problem has captivated me more than any other I’ve considered since, and I’m looking forward to hearing what you think.

Post your findings in the comments!


My newest artistic endeavor is jewelry–earrings and necklaces. See my Etsy shop for more.


Since the start of 2015, I’ve been interested in a simple mathematical flower structure, made of 52 regular pentagons. I’ve made this shape in a couple different sizes and media–the one pictured hereis 6 feet in diameter.Rose of Five Espresso Buzz

Couch Armada
I collaborated with an architect, a Boeing engineer, and a metalworker to build a full sized driving couch called Chassie. Powered by (2) 24V motors upcycled from mobility scooters. Controlled by a handheld joystick. Styled up with 4 speakers, a subwoofer, an aux input for audio, (2) 120V power outlets, 15m of LED lights, custom side pockets, and some rad orange trim.

Many thanks to Ignition Northwest for their financial support, the Seattle Design Festival and the EMP Museum’s Mini-Maker Faire for inviting us to come play, and to the many friends who helped along the way–particularly the Stroup family for helping with storage.

Hilbert Creations
I have aspirations to build a Hilbert-Cube-Jungle-Gym in the near future. If you’re interested in helping make this a reality, please let me know. Here is a very early prototype, made from PVC elbows (inspired by Chaim Goodman-Strauss).

PVC hilbert

My Pet Fractal
About a year and a half ago, I stumbled across this fractal. I recently had access to a laser cutter, and I made a few dozen handheld 256-gons. This curve is closely related to both the Sierpinski Curve and the Z-Order (Morton Order). If you know a name for this fractal, or if you’ve seen it in any literature, I would love to hear about it.

Fractal Tower

LED Array Art
This project is an excuse for me to improve my coding skills and a medium for me to share my mathematical musings more widely. I’ll be playing with variations of Conway’s Game of Life and other Cellular Automata, aiming to both create beautiful graphics and to inspire mathematical conversation in those around me.

Large Group Collaborations
Since 2013, I’ve played lead administrative roles in large art collectives of 25+ people. Each year, I’ve managed funds and budgeting for projects ranging from $5,000 to $30,000. In these roles, I’ve been paying particular attention to what it takes to bring to reality these large scale projects that sound so far-fetched.