Harmonic Leaning Tower (and Alignment Templates)

This is a STEM demo and learning game for the classroom.  In Round 1 you will search for solutions by guessing and checking.  In Round 2 you will follow an algorithm.  In Round 3 you will reflect on patterns and make an estimate.  Once you finish the game, be sure to check out the ridiculous size comparisons at the end.

The Big Question 

Start stacking blocks so that they lean over the edge of a table.  If you use one block per layer, how far can the tower lean?  How many blocks (or layers) does it take before the top block is completely past the edge?

Round 1: Subtracting

Print out 8-10 blocks (see details further below).  Then find a level surface and make a tower through trial and error.  Remember to keep the top block extended past the edge of the table.  Now take out one block and make a leaning tower again.  Continue taking out blocks until you find the minimum.  Are the overhangs at the top of the tower different than ones at the bottom?

Round 2: Adding

It may be easier to find the pattern by adding blocks instead of subtracting them.  Start with one block and make it lean over the edge of a table.  How far can it stick out? Half the length, or a little less to help it sit flat.

Now lift up the block and add another one on the table so that its edge is even with the table.  Shift both blocks together and move them further off the table without toppling the tower.  How far can the second block stick out?

When you find the maximum overhang for two blocks, lift them up and add a third block to the bottom.  Then shift the whole tower again.  How far can the third block stick out?  Keep adding blocks until the top block is completely past the edge of the table.

Round 3: Guessing

When you observe the overhang for each layer, do you see some sort of pattern?  Is this a pattern you have seen before?  Guess how many layers it takes for the top block to be completely past the edge by an additional block length.

Printing

To save filament while testing, I printed blocks with 2 perimeters, 3 top and bottom layers, and 20% concentric infill.  Be sure to choose a stiff filament because any bending will cause problems.  PLA is a great choice.

You need a perfectly symmetric infill pattern to make uniform blocks.  Several patterns look close, but concentric infill is guaranteed to be symmetric.  Because I was not confident in the strength and stiffness of this infill, I modeled some reinforcements.  You will see extra "internal" perimeters when you slice blocks. 

The edges of blocks should be very flat, so you need highly tuned first layer compensation (or elephant's foot compensation).  If extra material sticks out, the blocks will not work properly with the templates.  You can sand or file to remove the extra material, but try to avoid rounding edges.

Finding Level

Do your best to start with a level surface.  Instead of a tabletop, I used a flat scrap of wood and added 3 screws to the bottom.  Then I made fine adjustments to level by turning screws with a screwdriver.

To measure level, I used a smartphone app called Phyphox https://phyphox.org/.  Towards the bottom of the main interface is the Inclination tool.  I used it to measure level to two decimal places (0.00°).  You can use a bubble level, but this approach is much more precise.

If you are stuck with an uneven table, try rotating the direction of the lean until you find a stable orientation.  A tower that leans downhill may be difficult to balance.

Template Offsets

Super accurate and precise towers are fragile, on the edge of stability.  Blocks love to tilt instead of sitting flat, and the top block is usually the worst offender.  So, I added offsets to templates to gain extra stability.  (If you inspect every dimension, you will see that offsets are not evenly applied to each layer.  Layers towards the top get a larger offset, and the top block gets an additional offset.)

Offsets exist to simplify the process of making a tower, and to make the demo more reliable.

Which Template?

If you are reasonably careful, the medium offset should work.  If you are using blocks with small kids, try a large offset.  In my opinion, it is much simpler to start with a working tower and then nudge blocks further to test the limit of stability.  In contrast, it will be difficult and frustrating to start with an offset that is too small.

Fractions and Patterns

If you want to learn more details, there are plenty of guides that explain this problem.  Here is a Wikipedia article: https://en.wikipedia.org/wiki/Block-stacking_problem, and here is a Wolfram article: https://mathworld.wolfram.com/BookStackingProblem.html. 

A tower with maximum overhang is a visual and tactile example of a harmonic series.  Below is a table showing the maximum theoretical overhang for several layers in a tower.  Notice that each layer contributes to the total overhang. 

Do you see the pattern?  Overhang values for individual blocks are 1/(2 × layer number).  Or another way to say it: the bottom part of the fractions are 2, 4, 6, 8, 10, 12, ...

The fancy math notation on the cover photo gives instructions for generating this pattern of fractions.  The fancy "E" is the Greek letter sigma and means you should add terms together and compute a total value.  "i=1" and "n" count the number of layers, starting at 1 and ending with n.  Instead of listing any specific number, the placeholder n is used to describe a tower of any height.  The numbers (in this case fractions) after the sigma define the terms that are added together.  The placeholder i represents the current layer as you make fractions for each layer.

Answers

How many blocks does it take before the top one is completely past the edge?  Simply search the Total column for the first entry greater than (or equal to) one: 4 blocks.  With blocks that are 90 mm long, there is an extra overhang of 1/24 x 90 mm = 3.75 mm.  This is a small-ish target to hit, so do not worry too much if your answer was 5 blocks in Rounds 1 or 2.

The answer for two block lengths in Round 3 is 31 blocks.  Using the same 90 mm blocks leaves an extra overhang of 1.23 mm.

Infinite Overhang?

To infinity and beyond  ...  reasonable limits?  Let's circle back to the first question: How far can the tower lean?  The fractions keep getting smaller and smaller.  Is there a limit where the overhang stops growing?

According to the mathematical pattern, leaning towers can extend to any length.  However, the overhang grows very slowly and the number of layers quickly explodes.

I hope these size comparisons help you see why this concept can be difficult to fully realize in real life scenarios.

Thanks for playing along, and I hope you learned something!