# Physicist creates math model to predict maximum incremental domino size

(Phys.org)—J. M. J. van Leeuwen, a physicist at Leiden University in The Netherlands has created a mathematical model that predicts the maximum incremental size of falling dominos. He's found, as he describes in a paper he's uploaded to the preprint server arXiv, that in a perfect world, the maximum growth factor is approximately 2.

Most everyone has seen dominos in action. Small pitted black planks with white dots on them are placed on their ends next to one another – then at some point, the first is knocked over onto the second. The force of the first falling onto the second causes it to fall, knocking it down onto the third, etc. This continues until all the dominos have been knocked over without any other outside . Most domino exhibitions feature planks that are all of the same size, though most intuitively understand that different sizes could be used, which means a smaller domino can knock over one that is larger. But how much larger? That's the question Leeuwen posed to himself. He turned to math to find the answer and in so doing created a model that predicts not only how much larger a domino can be, but the chain length patterns that would occur using different growth factors.

Dominos fall the way they do because when one is stood on end, it possesses . That energy is released when it is pushed over. But because the force necessary to push the domino over is less than the amount of potential energy stored, it is able to knock over a nearby domino that is larger than it is, a known as force .

To create a , Leeuwen had to remove some real world factors that have an impact on chain reactions that occur when dominos are felled. Real dominos tend to slide at the bottom as they are knocked over, for example, and sometimes when one strikes another the result is an elastic collision that prevents the second domino from falling over. Also, sometimes dominos slide against one another as one strikes another. The result was a model that suggests the largest growth factor in a perfect world is 2, meaning one domino can knock over another that is twice its size.

The model also showed how quickly plank size can grow and still allow for a complete chain reaction. Starting with a plank just 10 millimeters high and assuming a of just 1.7, the model shows the planks growing to a size of the empire state building using just 244 planks.

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Domino Theory: Small steps can lead to big results

Abstract
The conditions are investigated under which a row of increasing dominoes is able to keep tumbling over. The analysis is restricted to the simplest case of frictionless dominoes that only can topple not slide. The model is scale invariant, i.e. dominoes and distance grow in size at a fixed rate, while keeping the aspect ratios of the dominoes constant. The maximal growth rate for which a domino effect exist is determined as a function of the mutual separation.

via Arxiv Blog

Journal information: arXiv

Feedback to editors

Jan 11, 2013
If the 2nd domino would be more than twice the size, the falling domino would tip it underneath the lower half of the domino, likely causing the 2nd domino to fall backwards.

My totally uneducated non-mathematical guess would be that the maximum growth factor of a domino would be around 2.

Jan 11, 2013
If you begin with a 10 mm "domino" and then increase by a factor of 1.7, repeating 244 times, then the last "domino" is much larger than the known universe. The last one will be 1.8e 38 lightyears in size. Which truly is larger than the Empire State Building.

TidyTim

Jan 11, 2013
The actual growth factor would need to be 4.5% to increase from 10 mm to 1453 feet in 244 increments. Using 1.7 you would surpass the height of the Empire State after 22 dominos (2,267 feet tall). [ CAUTION: dominos of in this size range can be dangerous :-) ]

TidyTim

Jan 11, 2013
Problems like these seem so simple - but if you've ever taken an university level mechanics class you'll know that even such trivially sounding problems are enormousley complex.
I'm not at all surprised that a lot of the complexity had to be taken out of it before making the calculations

Jan 11, 2013
Is this proof of the butterfly effect? :-)

Jan 11, 2013
As a Mechanical Engineer with enough credits for a few more engineering degrees this Kinematics problem is nice, but it's Degrees of Freedom and linkage structure isn't that complex. I guess the person had some spare time to kill.

Jan 12, 2013
The growth factor should depend on how thin the dominoes are relative to their heights. A really thin domino that barely stays balanced on its own can be knocked over by even a fairly small domino.

However the dominoes in the experiment look like they are only slightly thinner relative to their lengths than the dominoes in a box of dominoes, so the 'factor of two' limit is probably the perfect-world limit for the standard domino aspect ratio.

Jan 13, 2013
The potential energy explanation sounds a little glib to me.
What if you start with a large block and chip away at it until you end up with the dominoes in the starting position? Where is this "storing energy in the dominoes by putting them up on end" then?

Jan 13, 2013
@Manitou - even more potential energy is contained in the large block.
Some of the block's potential energy is removed with the chips.
The remaining potential energy when you have just the carved dominoes left is exactly the same as if you had stood them up individually.

Jan 14, 2013
@RealScience, I'm sorry, but this explanation is makes even less sense to me.

How is the potential energy greater if you carve vertical dominoes instead of horizontal ones?

By your explanation, cubic dominoes would have zero potential energy because they cannot fall down.

I guess my main objection to these energy explanations is that the objects themselves do not store the energy. The water in the city of Potosi does not have more potential energy at 4 km elevation than sea water because there is no system or machine in place by which it can fall 4 km.

Jan 14, 2013
Cubic dominoes have potential energy. They just don't have any NET potential energy between their initial state and their 'rolled over' state. Upright/elongated ones have such energy differentials.

The dominoes in the article have an energy difference between standing up and the lying down position (as can be easily seen by the potential energy formula m*g*h, where h is the height of the center of mass). That is the energy difference which can be transmitted to the next dominoe in the sequence

Actually the relevant force transmitted is less than that because it is a tradeoff between:
1) How far the domino has fallen. (The further it has fallen the more it has transferred ist potential energy into kinetic energy. More kinetic energy means more speed and hence more momentum imparted.)
2) Where the domino hits the other one. (the further up the less angualar momentum has to be transferred to make the other domino tilt out of its plane of stability)

Jan 14, 2013
Manitou - Let's start by clearing up what is meant by 'potential' energy. It means energy that could, under some circumstances, be released. (It has the potential to be released, whether or not it is practical to release it).

So water in Potosi DOES have more potential energy than sea-level water because if a machine were build it COULD fall the 4 km.

Returning to the dominoes, vertical dominoes have more potential energy than horizontal ones because their centers of mass are higher. Thus they have energy that could be released if their centers of mass were lowered.

Even a cubic domino has potential energy - you just can't extract any of that energy by ROLLING it because the center of mass ends up just as high.
However the potential energy is still there. Supposing the cubic domino were ice rather than wood - when it melts the water can still run down. Or if it is wood and you carve it, the wood chips fall down, releasing their potential energy.

I hope that this is clearer.