Domino Chain Amplifier
A line of dominoes where each is 1.5× larger than the last. Tap a 5 mm chip and watch the chain end in a slab that weighs more than you — a hundred-million-fold energy amplification, simulated with real rigid-body physics.
Now toppling
#1 of 13
Height
5.00 mm
Mass
0.0 µg
Energy released
0 J
Energy amplification vs domino #1
1.00×
0 of 13 down. A complete chain at ratio 1.50 multiplies the first domino's released energy by 2.83 × 10^8 (1.504 per step, compounded 12 times).
Chain log
| # | Height | Mass | Energy released | Amplification |
|---|---|---|---|---|
| Press Topple to tap the first domino. | ||||
Push the growth ratio past about 1.5×, or move the spacing outside roughly 0.7–0.9 × height, and the chain stalls. Finding that limit yourself is the point of the page.
Why the energy grows as the fourth power
Every domino in the chain is the same shape, just scaled up by a factor r (1.5 by default). That single fact drives everything. Two separate quantities grow at once, and they multiply.
Mass grows as r³. The domino gets taller, wider and deeper together, so its volume — and with a fixed material density, its mass — scales as the cube of the linear ratio. At r = 1.5 each domino is 1.5³ = 3.375× heavier than the one before it.
The centre-of-mass drop grows as r. When a domino of height h and thickness t falls flat, its centre of mass drops from h/2 to t/2. Both are proportional to h, so the drop is proportional to h — one factor of r.
Released potential energy is m · g · Δh, so it scales as r³ × r = r⁴. That is the whole trick. Each domino releases 1.5⁴ = 5.0625× the energy of its predecessor, and that factor compounds down the line.
m ∝ h³ (volume scales in all three dimensions)
Δh_com = h/2 − t/2 ∝ h (one linear dimension)
E = m·g·Δh_com ∝ h⁴ ∝ r⁴ per step
E_N / E_1 = r^(4(N−1))
The arithmetic, worked out honestly
Start with the simulator's defaults: a first domino 5 mm tall, thickness h/5, depth h/2, wood at 600 kg/m³, and a growth ratio of 1.5. Here is what a 13-domino chain actually produces.
| # | Height | Mass | Energy released | Amplification |
|---|---|---|---|---|
| 1 | 5.0 mm | 7.5 mg | 0.147 µJ | 1× |
| 4 | 1.7 cm | 288 mg | 19.1 µJ | 1.3 × 10² |
| 7 | 5.7 cm | 11 g | 2.48 mJ | 1.6 × 10⁴ |
| 10 | 19.2 cm | 426 g | 0.32 J | 2.1 × 10⁶ |
| 13 | 64.9 cm | 16.4 kg | 41.7 J | 2.83 × 10⁸ |
About that "two billion" figure. You will often see the domino amplifier quoted as roughly a two-billion-fold energy gain. That is correct — but it takes 14 dominoes, not 13. Thirteen dominoes means twelve growth steps, giving 1.5⁴ˣ¹² = 1.5⁴⁸ ≈ 2.8 × 10⁸. Add one more and you get 1.5⁵² ≈ 1.4 × 10⁹; a fifteenth takes you past 7 × 10⁹. The exponent counts the gaps between dominoes, not the dominoes themselves — an off-by-one that quietly moves the answer by a factor of five.
Whitehead's paper and the physical demonstrations
The canonical reference is Lorne A. Whitehead, "Domino 'chain reaction'," American Journal of Physics 51 (2), p. 182 (February 1983). Whitehead, then at the University of British Columbia, described a chain of thirteen dominoes each 1.5× the linear dimensions of the last, running from about 5 mm tall to roughly 65 cm. His point was pedagogical: this is the clearest tabletop illustration that a mechanical system can amplify energy by many orders of magnitude while every individual step remains completely ordinary.
Whitehead identified 1.5 as a reliable growth factor rather than a hard theoretical ceiling. He noted that the geometry permits a larger ratio in principle — analyses since have put the absolute limit closer to 2 for idealised rigid dominoes — but that in practice friction, imperfect alignment, bouncing and the finite stiffness of real wood eat into the margin. At 1.5 the chain works essentially every time, which is what makes it a demonstration rather than a stunt.
The idea has been built at full scale repeatedly. The best-known modern versions come out of the Netherlands, where researchers associated with the University of Twente staged domino-amplifier demonstrations for Dutch television and science-outreach events, using plywood and concrete slabs for the final members — the largest weighing on the order of a hundred kilograms and standing taller than a person. The physics is identical to the tabletop version; only the consequences of getting the spacing wrong change.
The real limit: it is a momentum problem, not an energy problem
A falling domino always has enough energy in principle to topple a much larger neighbour — that is what the r⁴ law says. The chain fails for a different reason: the energy has to get across the gap, through a single contact point, in the short time the striker is in contact. That is a question of angular momentum transfer, and it is far more fragile.
To topple, a domino only has to be pushed past its balance angle — the tilt at which its centre of mass crosses over the front bottom edge. For a slab of height h and thickness t that angle is θ_b = arctan(t/h), about 11.3° for the 5:1 proportions used here. Beyond θ_b gravity does the rest. Below it, gravity pushes back and the domino rocks onto its base again — which is exactly what a stalled chain looks like in the simulator.
- Too little spacing: the striker contacts the next domino after only a few degrees of rotation, when it has barely accelerated. It leans against its neighbour instead of hitting it, and the two settle into a static stack. Below about 0.5 × height the default chain will not propagate.
- Too much spacing: the striker rotates so far that its top corner arrives near ground level, striking low on the next domino's face. A low contact point is a short lever arm about the target's pivot, so almost no torque is delivered — and past roughly 1 × height the striker lands flat without reaching the next domino at all.
- Too large a growth ratio: the effective inertia the striker must overcome scales as r⁵ at the contact point while the momentum it can deliver scales far more slowly. Past about 1.55 in this model the strike simply cannot drive the neighbour through those 11 degrees, and the chain dies on the first transfer.
- The sweet spot: roughly 0.7–0.9 × height, which puts the impact high on the target's face while the striker is still moving quickly. This matches the spacing used in real demonstrations.
I·α = m·g·(h/2)·sin θ − m·g·(t/2)·cos θ
I = (m/3)·(h² + t²) slab about a bottom corner
θ_b = arctan(t/h) ≈ 11.3° balance angle for t = h/5
contact impulse: J = (1+e)·v_rel·(m_A·m_B)/(m_A + m_B), m_eff = I / r_contact²
What this is and is not a model of
The domino chain is the standard physical intuition pump for amplification: a small, precisely-placed input releasing a much larger store of energy that was already sitting there. Every domino was stood upright by someone; the tap does not create the energy, it only decides when the energy is released. That is the same structure as a transistor, where a tiny gate current gates a large channel current, or a relay, or the trigger on any stored- energy device.
It is a useful but imperfect analogy for a nuclear chain reaction. A domino chain is strictly linear: each domino topples exactly one successor, so the number of events grows arithmetically even as the energy grows geometrically. A true branching chain reaction — one fission releasing multiple neutrons, each capable of causing another fission — grows the event count exponentially, which is a structurally different and far faster process. The domino chain amplifies energy per event; a fission chain multiplies events. Conflating the two is the most common error in popular descriptions of both.
The simulation itself is a 2D rigid-body model: dominoes are perfect slabs rotating about a fixed front bottom edge, contact is a single-point horizontal impulse with a low coefficient of restitution, and there is no sliding, no bouncing, no deformation, and no friction at the base. Real dominoes slide a little, real contacts are distributed over a patch, and real chains sometimes propagate at ratios this model rejects. What the model does capture correctly is the part that matters: the r⁴ energy law, the balance-angle threshold, and the fact that the growth ratio and the spacing — not the energy budget — are what decide whether a chain lives or dies.