Math with LEGO
Lee De Cola
ldecola.net/projects/LEGO

  1. Bar plots
  2. Sine
  3. Circles and radii
  4. Data visualization
  5. Proofs without words
  6. Pi

This webpage illustrates some of the patterns I have made with LEGO. If you define mathematics as "the science of patterns" then LEGO are a tactile/visual computer to explore mathematics. Most of these experiments are in 2 dimensions, which seems quite enough space to get started!

Pile

When I opened the box I started to connect all of the 650 bricks in set #6177 in no particular order. As someone with a knowledge of discrete mathematics, I soon recognized that any connected "pile" of a finite number of bricks was one arrangement from a finite number of such arrangements, even though that number would be extremely large (probably beyond the number of particles in the universe).

Bar plots

3D bar plot I

All of the LEGO in the set are arranged by length × width × color. A true histogram uses the same symbology (usually same-width rectangles) to represent frequencies within same-with intervals, but this is a nice example of how objects can be used to represent their own frequencies.

This is an oblique view.

3D bar plot II

All of the 650 bricks in the #6177 LEGO set are arranged by length × width × color.

This is a plan view.

Grouped I

Here are all the bricks laid out by color × frequency × length × width. This is a useful way to organize the bricks out of the box to see exactly what you have, and it’s a natural way to illustrate hierarchical classification: dim1 × dim2 × dim3, with the first dimension changing most rapidly, etc.

Grouped II

Another way of arranging all the bricks that requires much more space. The cat is for scale.

Sine

Sine

This sine curve was my first attempt at illustrating mathematics with LEGO. At first I merely computed the rounded values of the sine function at every 10 degrees (spanning 36 units) but this left gaps in the function, which I filled in by computing the inverse, arcsine() function.

The exercise helps you to understand the problem of "aliasing" in computer graphics.

Sine and cosine

As you add bricks to the sine representation you acquire a nice sense of the rhythms of the pattern: first you realize that the arrangement repeats every cycle, then (in a negative direction) every half-cycle, then as a mirror image every quarter-cycle.

Multi-phase

I continued with the sine curve in multiple phases, trying to fill up the space. I think I ran out of enough distinct colors, but I’m sure that finishing this − and extending it in the horizontal direction − would make a pleasing pattern. There are only 9 colors in the set I used, but I think there are a couple of dozen more.

Double helix

A variation on the sine function is the addition of another sine with 2-color connectors every 4 units. The use of only 4 colors for the connectors represents the 4 amino acids.

Circles and radii

A circle

A circle is defined by r2 = x2 + y2 or y = sqrt(r2x2).

Here’s circle with radius 40. As with the sine curve, you have to compute the value of the function until its derivative is < −1 (in brick-space), then you need the inverse.

Part of a circle of infinite radius is shown in white.

Circles I

To plot circles of radius = 0, 1, ⋯., 22 (when I ran out of orange), I wrote an R script that did the following:
  • compute the rounded values of y for r = 1, 2, ⋯, n / √2 because you only need values up to the line of equality y = x,
  • determine the run length encodings for each r,
  • print out only lengths of each run.
Here are a few results:

  18: 5 3 2 1 1 1
  19: 5 3 2 1 2 1
  20: 5 3 2 2 1 1 1
  21: 5 3 2 2 1 2 1
  22: 5 3 3 1 2 1 1

Radii

For each angle θ in the set {i × π/32 | i = 0, 1, ⋯, 8} compute for each x = 0, 1, ⋯, 48 the value y = round(x tan(θ)). (The gray base board is 48 bricks wide.)

These brick lengths can also be summarized with run length encoding. Here’s the result for angle #6 (blue):

  COUNT: 1 7 1 6 1 7 1 2
  BRICK: 1 2 1 2 1 2 1 2

Start with a 1-brick, then seven 2-bricks moving up each time, then one 1-brick, six 2-bricks, etc. For angles #8 and #9 I ran out of 1×1-bricks. And you might think from examining the ‘lime’ bricks that tan(3/16 π) = 2/3, but at x = 112 units you’d encounter a break in the 1-2 pattern.

Circles II

Cells are at increasing distances from a central cell, here each distance is shown with a different color. I should add to this model, which I think requires only 1 × 1 bricks.

Data visualization

Time series

This visualization plots yearly global temperature anomalies from NASA. The horizontal axis spans the years 1970 to 2015 with 1×1 and 1×2 bricks marking 5 and 10 years respectively. The vertical axis spans -0.25 to 1.00 °C with bricks marking 0.25 °C increments. The overall trend is upward, but with an apparent recent leveling off. Nevertheless a regression fit to the data suggest a .02 °C/year increase, or over 3 degrees °F per century.

The design is somewhat busy; the ‘buttons’ work well, but the points should be of a greater contrast, and the axes less.

A 22 × 16 grid

This simple ‘map’ was derived from the volcano (Maunga Whau, New Zealand) digital elevation data that comes with the R language (2014). I sampled every 4 rows and columns (so the unit brick = about 40 meters) and collapsed the elevation values to 9 classes. The colors were selected from those available to suggest an island with a brown shoreline, lowland vegetation, arid uplands, and a lake surrounded by snow.

Spatial autocorrelation gives the image a sense of being about something real; to me the image could also be a false-color micrograph of a bacteria or perhaps a supernova⋯

Cracking problem

During Feb-Apr 2014 while I developed the various patterns shown above I began to notice cracks in the bricks. Overall, 257 or 40% of the bricks from the 650 in the set #6177 that I originally purchased showed some signs of cracking on at least one (and occasionally up to 4) sides. All but 2 of the cracks appeared in the width-1 bricks, for which a detailed analysis of the incidence of failure is summarized here with representation in which colors and lengths of width-1 bricks showed cracks. The crosstab shows the 4 lengths by the 9 colors and the 5 colors on the right represent incidence with breaks at 20, 40, 60, and 80%. The image shows that the 1 × 1 bricks fail most frequently, as do the blue and lime, but of course this is not a scientifically, controlled experiment.

Seven types of brick color × length types show cracking in all the items of that class. Although I was given a new set of bricks, I have not yet received a satisfactory answer from Lego.

Proofs without words

I was introduced to Nelsen (1993) by Lynn Salvo of MathCamp. It’s a wealth of ideas about how to convey mathematical ideas using a purely visual language. Unit Lego bricks, their grid organization, and a range of colors are all you need to get started.

Sums of integers I

This exhibit is step 6 in the theorem that
    1 + 2 + ⋯ + n = 1/2 n (n + 1).

Sums of integers III

Although it’s not in Helsen (1993) this exhibit also illustrates step 8 of the above theorem.

Sums of odd integers I

This exhibit shows step 8 of the illustration that
     1 + 3 + ⋯ + (2n − 1) = n2.

Sums of odd integers II

Step 5 of the illustration that
     1 + 3 + ⋯ + (2n − 1) = 1/4 (2n)2 = n2.

Squares and sums of integers

Step 6 of the illustration that
    1 + 2 + ⋯ + (n − 1) + n + (n − 1) + ⋯ + 2 + 1 = n2.

Pi

Digits of pi

For Pi Day my wife Freya and I arrayed 120 digits of pi using a different color for each digit. The design also illustrates how to make letters on a 5 by 8 grid.

Materials for pi

Here are all the bricks for the previous exhibit (including the title) arranged by color length × color × width. This exhibit illustrates the process either of categorizing the objects that make up one pattern, or of assembling the materials to create a pattern.

A Pythagorean triangle

The 3-4-5 Pythagorean triangle is here doubled. You can’t do this in the ‘two dimensions’ of a single brick layer.

References

Nelsen, Roger B (1993) Proofs without words: exercises in visual thinking. Washington, D.C., The Mathematical Association of America.

R Core Team (2014) R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. www.R-project.org