A Recursive Process

Note: this post is a more advanced version of the previous post and uses slightly different software. All this software is free and open-source.
In this post I’m going to outline how to use a workflow to go from a desmos sketch to a processing sketch to Fiji to a 3D file. The basic idea of the workflow is based on the idea of looking at a 3D object as a 2D image which is changed through time. Here’s a gif (source) that shows how we can view a cube (a 3D object) as a 2D slice.

So if we can create the 2D sketch, then we can create the 3D object with Fiji. Here’s a video that contains a walkthrough of all the steps involved in this process in more detail.

Desmos Sketch

I used Desmos as a tool for quickly prototyping the 2D sketch. The original idea was to make a wavy cup. So the cup would start with a circle base, and slowly change up the walls to have a wavy top. Here’s the desmos sketch that shows the cup being sliced from the bottom to the top:

(There is more detail on how this sketch was created in the video.)

Processing Sketch

Next, we can bring the sketch to Processing so that we can easily save a bunch of frames and have a lot of control over all the details. Here’s the live code (had to be modified for openprocessing because PShape isn’t supported in JS), and here’s the original code. Once again, much more detail is available in the video.

Fiji

Next bring the 400 frames into Fiji. Fiji is often used for stitching 2D images from MRI machines into 3D objects.

MeshLab

This is an optional step for shrinking the number of triangles used, and hence shrinking the model size.

3D Print

Shown with Makerbot, but every 3D printer has software that can do these steps.

and two hours later:

More Wavy Cups

These both use a different period for the sine function, and the second adds in a sine inside the first sine based on the height to give it the “wiggle” back and forth.

Spiral Ball

These were teased in a couple of tweets, the concept is the same as the wavy cup, but the size of each slice is controlled by a circle function. The desmos sketches are linked in the tweets.

If this represents slices of an object over time then what does the 3D object look like? https://t.co/PPjUGzvWj3 pic.twitter.com/btLQoaZuWp

— Dan Anderson (@dandersod) June 30, 2016

Likewise: https://t.co/dy4UdzTKzd pic.twitter.com/j3TocBT3iF

— Dan Anderson (@dandersod) June 30, 2016

The black spiral ball uses consecutive fibonacci periods spinning in different directions, which is why it looks like a pine cone.

Fresnel Lens Sphere

Here’s the desmos sketch for this (admittedly weird model).

Dragon Fractal

These were directly coded in processing, here’s the originating code to make the fractal.

Let me know if you know of a different way of making this kind of 3d model! It works pretty well, but there are some rough edges with taking the 2D images to make the 3D model. Cheers!

The seniors don’t go to school on the last day, so I had calculus class on Friday with one student (a junior). We went down to the innovations lab to create things. Here’s what we worked on:

Took this sketch from dailydesmos:

I had the student work on the solution after talking about superellipses (or the better name, squircles).

When he finished the solution, we modified it to get this sketch:

and we decided to try and get a 3D print of this sketch. We started working in madeup (a fantastic beta of a program that takes code and creates 3D models), but found it tricky to graph the function. Went back to desmos to get this sketch where we could figure out how to get the specific coordinates of points along the curve.

Fiddled around for a bit to get this result:

Here’s the code:

to abs x = if x > 0 then x else -x

to func t_ a_
  if (t_ < 0)
    t_ = t_ * -1
  end
  out = a_ - t_ ^ 4
  if (out < 0)
    out = abs out
    out = out ^ (0.2)
    out = - out
  else 
    out = out ^ (0.2)
  end
  out
end


a = 20
h = 0
while a > 0
    t = -3
    while t < 3
        x = t
        y = func t,a
        moveto x,y,h
        t = t + 0.025
    end
    h = h + 0.1
    extrude 0,0,1,-0.1
    a = a - 1
end

Then click on the solidify button and download the model.

Send to makerbot, and pick up first thing monday!

I’ve been interested in the intersection between Math and Programming ever since I taught my first programming course 12 years ago (call it coding, CS, whatever you’d like). I think I’m finally starting to figure it out. Maybe. I was prodded by Dan Meyer and Julia Tsygan to think more in detail about this intersection:

Hi! I do a whole bunch of math in programming class. Only a small bit of code in math class. Darn curriculum.

— Dan Anderson (@dandersod) June 11, 2016

huh, so are you saying this is uncharted? what code in math class do you do? What math in programming and why?

— Julia Tsygan (@JuliaTsygan) June 11, 2016

we do a whole Mandelbrot fractal unit in precalc where we teach complex numbers through the fractal.+

— Dan Anderson (@dandersod) June 11, 2016

I show the code I made and they help fill in the missing pieces from the complex number math.

— Dan Anderson (@dandersod) June 11, 2016

Failures

I’ve failed at teaching math topics directly through programming in math class. Maybe it’s a symptom of teaching math courses that are CHOCK full of material (IB PreCalculus Honors, and IB Calculus HL), but I just don’t seem to have the time to have the students work through the material at an appropriate rate. Or I’ve picked bad topics to try and teach through programming. Or it’s been a challenge to adapt to the different levels of student programming knowledge coming into math class. I don’t know. I’ve tried teaching arithmetic and geometric sequences and series with Python and loops, and I’ve tried teaching some probability through Python. No luck. I think it boiled down to the following problem:

• It’s tough to teach programming concepts AND sequences and series without taking a long time to build up. How can you (quickly) learn whether or not a series converges without having first understanding how a loop works, how variables work, and maybe how conditionals work. For example, this line of code, while easy for a programmer to understand, is *really* confusing for a new programmer in math class

  x = x + 1

  This line of code takes the current value of x, adds one, then stores it back into the variable x. “=” in most programming languages is an assignment operator, and “==” is an equality operator. Ich.

Small Successes

I think there ARE ways to teach a bit of math through programming. I just think that it’s use is limited. I presented on one of these topics at NCTM’s Annual Meeting in April of 2016, and at Twitter Math Camp in July 2015. All my resources are found here: Geometry from Scratch. Click through to get more details, but it boils down to the idea that you can be successful teaching Geometry or Middle School students about angles and polygons by having them use focused play in Scratch. They’ll discover the Exterior Angle Theorem after they’re asked to draw a hexagon. Likewise you can teach Geometry, Algebra, or Middle School students about slopes and lines by having them draw lines by making stairs.

Can you teach a big part of the math curriculum through programming in math class? Sure! In the same amount of class time? Nope!

Larger Successes

I think the biggest gain can be found by teaching math topics in programming classes. The sticking point? They might all learn DIFFERENT math topics. At the end of our programming class, I give the students about 4 weeks to work on whatever project that they’d like. It’s graded on a loose rubric that is just looking for programming milestones. Out of a class of 16 students this year, here is a subset of their work when it came to learning and applying math topics to their work. Keep in mind, I gave no guidance to them learning math topics, these came up naturally from our previous work, and after they saw some examples of mathematical computer art (more on this later? Future blog post?)

(This is my student math art gallery that just started!)

The left picture is from a student who was having trouble landing on a specific topic and so just ended up making many smaller ideas into one project. You can read more in her blog post.

Here’s some code from her (all completely from scratch):

 while (theta < 4*PI) {
    x = r*cos(theta);
    y = r*sin(theta);
    z = 2*r*cos((theta+PI/25));//finds the new point that the line should go to
    s = 2*r*sin((theta+PI/25));
    x = map(x, -400, 400, 0, width);
    y = map(y, -400, 400, 0, width);
    z = x-z;
    s = y-s;
    lines(x,y,z,s);//calls the function lines
    theta = theta + PI/25;//repeats it until theta = 4PI so that the lines go in a circle
    }

This student is using polar coordinates that she learned in PreCalculus class, trigonometry with radians, and has used a proprortion to work with the colors (the map command).  Later on the student uses the % operator, which calculates the remainder so that she can cycle through a set of drawing commands.

The picture on the right is from a student who was going to extend the Geometry from Scratch polygon angles from above. But the move and turn commands were too limited for him, so after a 5 minute talk, he learned how to use polar form of coordinates to easily make a n-gon from a given center. Keep in mind, that this student hadn’t heard of polar coordinates before and mastered them to get these shapes. I’m excited to see how much knowledge he’ll bring to PreCalculus next year when polar coordinates are officially “taught” to him. Here’s his blog post for his project.

This student designed a 3d model with code to create this fantastic piece of math art. It is a series of cocentric circles that represents the digits of pi, the center of the circle has a small circle with height 3. The next circle has height 1, then 4, then 1, … Here’s his blog post with code samples. He used 3D polar coordinates as well, and he made many connections to his project and the integration unit of his calculus class (rotational volumes).

This graphics program uses polar coordinates (yep again!), transparency, circles, trigonometry, lines, triangles and so much more to make some fantastic images. His blog post with more info.

This student did a project that has a grid of rectangles in a 3D environment and he rotates them in 3D based on a periodic pattern and the mouse position. Some really great use of translations, rotations, and 3D geometry. Here’s his blog post.

This student made a tank shooter game where the user (blue) battles the AI enemy (red). The really great math that this student had to apply and learn was the use of arctan to get an angle between two coordinates so that he could shoot between the coordinates. Impressive math. Blog post (sadly without the code or a link!).

Lastly, here’s a student’s work on making a bunch of bouncing coins. The math that this student had to work through was using vectors to make a gravity system and to obey Newton’s first law of motion. This senior hadn’t taken any math class that had vectors, nor had he taken any physics class. Yet he mastered the idea of velocity vectors to get the motion to work properly. He also was somewhat successful in making a collision system (from scratch!), a very hard task. His blog post.

Conclusion

I don’t know. That feels like much of my progress into adulthood. I’m less definitive about statements. I know that I don’t know a whole lot.