Tumbling Cube and the Action of the Mind

Scientific Paper
Title	Tumbling Cube and the Action of the Mind
Read in full	Link to paper
Author(s)	Jean E Burns
Keywords	Tumbling cubes, Mental influence, Quantum randomness, Uncertainty principle, Gauss
Published	2002
Journal	Noetic Journal
Volume	3
Number	4
No. of pages	15
Pages	318-329

Read the full paper here

Abstract

It is shown that a very small change in the initial angle of orientation of a tumbling cube can be detected by a shift in the endpoint of the trajectory after the cube travels a minimum distance. Specifically, if a cube travels forward in the x direction, with s the number of forward steps (tumbles) in the trajectory, the average sideways deviation DY in final position produced by a change in initial angle 0 dq is given by DY = a(s ? s₀), where a is the average length of the sideways step during each tumble and s₀ a term which depends logarithmically on 0 dq . It is proposed that the reason cubes (dice) are popular in games of chance is because this ability of the cube trajectory to magnify small changes allows the possibility for mental influence to act. A general outline is given for experiments using traveling cubes which can test for the action of mental influence.

Overview

This is a mechanics paper written in the service of a parapsychological question. Jean E. Burns asks how large a macroscopic effect a very small perturbation applied at the start of a die's trajectory can produce by the time the die comes to rest, and answers with a closed-form estimate: the sideways displacement grows linearly with the number of tumbles once a threshold number of tumbles has been passed, and the threshold depends only logarithmically on the size of the initial perturbation. The point of the exercise is that an influence too small to observe directly — Burns explicitly declines to specify its magnitude, assuming only that it is small — can nonetheless be detected at the end of a long enough trajectory.

The framing acknowledges from the outset that the existence of mental influence is unsettled. Burns states plainly that "it is not presently known whether free will or any action of the mind exists," that brain neurophysiology is not understood well enough to settle the question by direct measurement, and that presently known physical laws "encompass only determinism and quantum randomness," so that a genuine volitional process would require "a radical addition to these laws." The paper builds on Evan Harris Walker's 1975 observation that a travelling cube is extremely sensitive to small changes in initial angular orientation, but replaces Walker's rough treatment with a more detailed dynamical analysis and, importantly, detaches it from Walker's specific claim that mental influence operates within the limits of the uncertainty principle. What departs from mainstream practice is not the mechanics — which is ordinary rigid-body dynamics — but the proposed application: designing a psychokinesis experiment around an amplification mechanism.

The argument

Trajectory counting and the threshold s₀

A cube travels in the x direction, either by tumbling (a corner always in contact with the surface) or by bouncing. When tumbling, a corner is held by friction while forward momentum rotates the cube about it; gravity meanwhile torques the centre of mass sideways, so each forward step is accompanied by a sideways step of average length a, to the left or right according to which side of the contact corner the centre of mass lies.

Burns approximates the continuum of possible trajectories by a discrete binary tree: after s forward steps there are 2^s distinct left/right sequences, each assumed equally probable. The initial angle θ₀ between the corner-to-centre-of-mass vector and the vertical ranges over π/3 by the cube's geometry, so each discrete trajectory corresponds to an angular width (π/3)/2^s. A perturbation δθ₀ therefore spans m = 2^sδθ₀/(π/3) trajectories.

The counting argument is elegant. If δθ₀ is just large enough to flip the last step, two trajectories are in play, separated by 2a, and the average shift is a. If it flips the last two steps, four trajectories span 4a and the average shift is 2a. In general a shift of ra requires 2^r available trajectories, so ΔY = a log₂m, which reduces to

ΔY = a(s − s₀), s₀ = log₂[(π/3)/δθ₀].

The interpretation of s₀ is that it is the trajectory length at which δθ₀ exactly fills the angular width of a single discrete trajectory: below it the perturbation cannot move the cube onto a neighbouring path at all. Burns then shows that if the influence acts continuously rather than only at the start, the extra contribution from later steps forms a geometric series summing to less than a doubling — equivalent to reducing s₀ by at most one step. Almost all of the deviation is therefore produced in the first few tumbles, while the magnification requires a long trajectory. Because s₀ is logarithmic in δθ₀, the dependence on every factor contributing to the initial perturbation is logarithmic too.

Step lengths and the parameter ζ

To use the formula one needs a and the forward step Δx. Burns notes these are hard to measure — high-speed film cannot reliably show which corner the cube is on — and derives them instead. With b the half-length of a side and the corner a distance √3b from the centre of mass, the parallel-axis theorem gives the moments of inertia about a corner as I_φ = I_θ = 4γMb² with γ near unity. A skew-oriented cube tumbles about six corners per full rotation, so the average forward rotation per tumble is 60°.

The sideways rotation obeys d²θ/dt² = Cθ with C = √3Mgb cos i/I_θ (i the ramp inclination), whose growing solution is θ = θ₀exp(C^1/2t). Burns defines a dimensionless parameter ζ by requiring ζt_φ to be the time for θ to grow from its mean value π/12 to the maximum π/6, obtaining ζ = u/[6.62b^1/2(cos i)^1/2] in cm^1/2/sec units. If ζ > 1 the next corner arrives because of forward rotation; if ζ < 1 it arrives because the sideways rotation has saturated. The convenient result is that for ζ = 1 the two step lengths lock into a fixed ratio: Δx = √3b and Δx/a = 4, so that

4ΔY = X − √3b s₀.

Written this way, a plot of (Δx/a)ΔY against forward distance X rises at 45° and the intercept reads off X₀ = √3bs₀ directly.

Target faces, Gaussian envelopes, and bouncing

For a gambler the relevant question is not lateral drift but which face lands up. Since six corners span a full rotation, a sideways rotation of ±π/3 makes two further faces accessible; with Δθ = π/12 at ζ ≈ 1, only f = 4 steps past s₀ are needed. Burns is careful not to convert this into a probability of 1/2, for two stated reasons: which face ends up depends on how the cube stops (friction, rolling on an edge, or a dead stop when two corners land together), not on lateral deviation alone; and random fluctuations blur the outcome.

Those fluctuations get their own treatment. With all 2^s trajectories equally weighted the endpoint distribution is Gaussian with standard deviation s^1/2a, and air currents and surface irregularities add further randomness. The observable ΔY is therefore the separation between the midpoints of two Gaussian envelopes — one with intention, one without — and the signal-to-noise ratio is ΔY/⟨Y²⟩^1/2 = (s − s₀)/s^1/2, so many trials are required. The discrete-trajectory idealisation also makes ΔY vanish identically below s₀; since the real trajectory set is a continuum, Burns notes the true curve rises smoothly from zero, and steepens near the end of travel as the forward step shrinks faster than the sideways one.

The Section 2 derivation, she argues, transfers unchanged to bouncing cubes — all it requires is s forward steps with an average sideways step a and equal weighting of the 2^s paths. But bouncing cubes have longer forward steps, so a given distance contains fewer steps, and since s₀ is unchanged the cube must travel further before amplification begins. Bouncing cubes also acquire their sideways motion from collisions rather than gravitational torque, at the cost of forward kinetic energy.

An estimate from Forwald's data

Burns applies the model to Haakon Forwald's cube-deviation experiments. Cubes with b = 0.8 cm travelled about 15 cm down a ramp plus 35 cm across a table, total X = 50 cm, with typical deviation ΔY ≈ 5 cm. From ζ = u/55.6 and Forwald's stated u₀ = 186 cm/sec at the tabletop, ζ exceeded 3 at the start, so Δx ≈ √3b and s ≈ 36. Taking Δx/a ≈ 2 (raised from 4 because the cubes struck one another on the ramp), with a range 1.5–2.5, gives s₀ = 29 ± 2, or X₀ = 40 ± 3 cm — and ΔY/⟨Y²⟩^1/2 = (36 − 29)/36^1/2 = 1.17, so the claimed shift was about one standard deviation of the underlying spread.

The experimental recommendations follow from the analysis: shield the cubes from breath and hand-driven air currents (Burns notes pointedly that earlier parapsychology work did not, and that Forwald stood beside the table); use a gentle ramp to hold ζ near 1 and keep the cubes tumbling; ensure enough friction that cubes tumble rather than slide; alternate right-intention and left-intention blocks and take half their difference, as Forwald did; and do not let multiple cubes strike each other if the ζ ≈ 1 relation is being relied on.

Assessment

The mechanical core of the paper is its strength, and it is largely independent of the parapsychology. The binary-tree argument giving ΔY = a(s − s₀) with logarithmic s₀ is a clean, physically transparent statement of exponential sensitivity to initial conditions in a dissipative tumbling system, and the demonstration that a continuing influence adds at most one step of amplification — because the number of reachable trajectories decays geometrically — is a genuinely useful result that is easy to get wrong intuitively. The ζ parameter and the resulting Δx/a = 4 lock-in give experimenters a concrete design target rather than a hand-wave. Most creditable is the paper's epistemic hygiene: Burns repeatedly says the existence of mental influence is unknown, treats it as a hypothesis to be tested rather than a premise, insists that the deviation must be extracted as a difference between two Gaussian midpoints, and warns that random air currents will be amplified by exactly the same mechanism — an argument that cuts against her own hypothesis and that she nonetheless makes prominently. Her identification of unshielded breath and hand motion as a confound in prior dice work is a criticism of the parapsychological literature from inside it.

The difficulties are of two kinds. Within the mechanics, several steps are asserted or estimated rather than derived. The assumption that all 2^s left/right sequences are equally probable is flagged by Burns herself as possibly not strictly true, yet the entire counting argument rests on it; a real tumbling cube's left/right choices are correlated through θ, not independent coin flips. The moments of inertia are approximated by assuming the centre of mass sits directly above the contact corner and that the two centre-of-mass moments are equal, with a fudge factor γ "close to one" that is then dropped. The linearisation sinθ = θ and the substitution of a fixed average cosθ = cos(π/3) are reasonable but uncontrolled. The relation Δx/a = 4 holds only at ζ = 1, and in the one case actually worked out — Forwald's — ζ was above 3 and the ratio was chosen as 2 on qualitative grounds. Since s₀ is obtained by subtracting (Δx/a)ΔY from X, that choice largely determines the answer; the quoted s₀ = 29 ± 2 carries an uncertainty from the assumed ratio range alone, and does not include the uncertainty in s, in the average Δx over a mixed bounce-and-tumble path, or in Forwald's deviations themselves.

The deeper problem is the inference the amplification is asked to support. The paper establishes that a small initial angular perturbation is amplified — but this is a property of any small perturbation, including exactly the air currents, table vibrations, release-mechanism variations and experimenter-correlated draughts that Burns lists. Sensitivity to initial conditions is symmetric: it makes a psychokinetic signal detectable in principle, and it makes every mundane confound equally detectable, while amplification adds no evidential weight favouring the former over the latter. The empirical anchor is also weak. Forwald's results are reported as significant but the paper concedes he took no precautions against breath or hand motion, which by the paper's own analysis is the single most dangerous confound; using them to fix s₀ therefore risks calibrating the model on an artefact. The Radin and Ferrari meta-analysis of dice experiments is cited as showing a statistically significant effect, but such meta-analyses have been contested on grounds of selective reporting and small-study bias, and the paper does not engage with that literature. Finally, the physical mechanism is left entirely open: Burns is candid that no known law provides for volitional influence on matter and that "a radical addition" would be required, so the paper is best read not as evidence for psychokinesis but as a careful specification of what an experiment would have to look like — travel distance past s₀, ramp geometry near ζ = 1, sealed against air currents, balanced right/left intention blocks, large N — before its result could mean anything. On that narrower aim the paper is sound.