# The Leaning Tower of Lire

May 04, 2020
Hello. My name is Michael and in this episode of Michael's Toys we are going to play with: Blocks. I'm sure you've played with blocks before and noticed that it's quite fun to put one block on top of another. It keeps. You can continue doing this and build a

#### tower

as tall as you want. But what if you don't want the

#### tower

to just rise? What if you also want the tower to go to one side? How far can you go to the side... ...without falling? Well, this question is known as the block stacking problem and its solution is The Leaning Tower of Lire.
In fact, you can mechanically build a Leaning Tower of Lire, simply by touch, taking several blocks. Here I have five, and notice that when I place one block on top of another, that top block can be pushed outward... ...but only up to a certain point, beyond which its center of gravity: the point from which gravity appears be pulling it down; is no longer above the support, torque is produced and the object rotates. So if I make sure the center of gravity is right above the stand, it will stay. But now I can treat both blocks as a single object and balance them on a third block.

## the leaning tower of lire...

Now, just by feeling - without using math or engineering - I'm going to see how far... ...both blocks... ...can stick out from this third bottom. Well, okay, that's too much, but you can rebuild. Oh. Perfect. Fourth block. Well, it's not really heavy; I am very weak. Well. Now, this is... Can you go further? No, it's about... Wow, okay, the fifth block. Here we go, fifth block. Again, I'm taking this to the limit, to the extreme, every step of the way... ...but it's hard, because of course; I'm doing this... ...in real life. Now let's see... ...if... Well, this one needs to come in...
Good! So we have built here the beginning of a Leaning Tower of Lire: I say beginning, because this tower will have no end. You can keep doing this forever; and your tower of individual blocks can extend to one side as much as you like: but there are diminishing returns, because the amount of overhang we get with each new block decreases, and it decreases by a specific amount. Now, I did this with blocks that aren't really perfect: they have holes, they're not completely homogeneous, and I'm not very good at balancing things; but if you look at it, if you look at the spaces closely: you'll notice that here at the top, the top block may stick out from the second block by about half its length, but then the second block sticks out from the third. by about 1/4, and then we have 1/6, and then we have 1/8. 1/2, 1/4, 1/6, 1/8...
Then would come 1/10, 1/12, 1/14, 1/16... This is a Harmonic Series. The numbers: the amount of overhang, gets smaller and smaller with each new block we add. In fact, it turns out to be 1/(2n), where n is the number of blocks. Here we have 4 blocks and the overhang is 1/(2n). 2 * 4 = 8, so it is 1/8. BUT; Although the amount of protrusion we can get is getting smaller and smaller, it never reaches 0. Therefore, these blocks can protrude as much as we want; as long as we have enough. I pitched this concept to Adam Savage and in his workshop we built a Leaning Lire Tower... ...with more than five blocks.
ADAM: Michael, you want to build something or prove something. MICHAEL: I want to build a Leaning Lire Tower. ADAM: A

### leaning

tower of Lire? MICHAEL: An inclination, yes. MICHAEL: It's all about the hangover. MICHAEL: No, not the bad kind, but the interesting kind. ADAM: Not the bad kind, yeah. Well. MICHAEL: I have a playing card here. ADAM: Yes. MICHAEL: And it's pretty obvious that it will balance on its center of mass, right? ADAM: Mmm, yes. MICHAEL: Well, I can hang the card on a table... MICHAEL: ...by lining it up so that exactly half of the card is off the table and the other half is on top.
MICHAEL: Because each next ledge is smaller than the last... MICHAEL: Because each next ledge is smaller than the last... ADAM: Yes. I see. MICHAEL: ...and the order is simply 1/2, 1/4, 1/6, 1/8, 1/10, 1/12,... MICHAEL: Playing cards are great because they are so thin that, already You know, 31 of them aren't even as thick as a deck of cards... MICHAEL: ...and 200 of them are only 4 decks. ADAM: 4 decks, yes. MICHAEL: So it's actually not that high; but they're also usually constructed with this type of air cushion... MICHAEL: So it's actually not that high; but they are also usually constructed with this type of air cushion...