Mathematica is often thought of as a sophisticated calculator, graphing program, or teaching tool.
But in mathematics and theoretical physics it has also served as something closer to a laboratory:
a place where researchers can experiment numerically, manipulate exact symbolic expressions,
visualize complicated systems, spot unexpected patterns, formulate conjectures, and sometimes
discover results that would have been extremely difficult to find by hand.
It is worth making one distinction. Not every example below was discovered exclusively with
Mathematica. Some involved Maple, PARI/GP, custom programs, or specialized software. But all
illustrate the style of experimental mathematics that Mathematica helped make practical:
compute → recognize a pattern → conjecture → prove.
1. Discovering remarkable formulas for π
One of the most famous examples of experimental mathematics is the 1995
Bailey–Borwein–Plouffe formula:
π =
Σk=0∞
16−k
[4/(8k+1) − 2/(8k+4) − 1/(8k+5) − 1/(8k+6)].
Its extraordinary consequence is that one can calculate a hexadecimal digit of π far out in
its expansion without first calculating all the preceding digits. High-precision computation
and integer-relation algorithms such as PSLQ were crucial to discovering formulas of this kind.
This illustrates a striking modern technique: calculate a constant to hundreds of digits, ask
the computer whether those digits conceal a simple relation involving known constants, and then
attempt to prove the resulting conjecture.
2. A mysterious harmonic-number identity
As an undergraduate, Enrico Au-Yeung numerically investigated the series
Σn=1∞ Hn2/n2
and conjectured that it equaled
17π4/360.
Jonathan Borwein initially suspected that the agreement was accidental. Using high-precision
numerical integration with systems including Mathematica and Maple, the identity was checked to
many more digits. It was correct and helped stimulate further work on what are now called Euler
sums.
This is an especially clean example of a computer turning a weak numerical hint into a
sufficiently convincing conjecture to justify looking for a proof.
3. New identities involving the Riemann zeta function
Researchers have used hundreds of digits of numerical precision together with integer-relation
algorithms to discover unexpected formulas involving values such as ζ(3), ζ(5), and
related sums.
Some discoveries generalized the remarkable series that appear in Apéry's proof that
ζ(3) is irrational. Instead of starting with an elegant identity and checking it on a
computer, researchers sometimes began with a mysterious decimal number and let computation
suggest the exact expression hiding behind it.
4. Unexpected links between particle physics and knots
Very complicated Feynman-diagram calculations in quantum field theory produced numerical
constants whose structure was initially obscure. High-precision computation and integer-relation
searches helped reveal connections among Feynman diagrams, multiple zeta values, and knot theory.
That is a remarkable conceptual jump: an integral describing particle interactions can contain
mathematical structure associated with knots.
5. Exact formulas hidden inside statistical-physics integrals
Researchers including David Bailey, Jonathan and Peter Borwein, and Richard Crandall evaluated
complicated integrals from statistical physics to hundreds of digits.
Numbers that looked like arbitrary decimals turned out to have exact forms involving familiar
mathematical objects such as zeta values and Dirichlet L-functions. Computation also exposed
unexpected recurrence relations, some of which were later proved.
6. Mathematica in black-hole and gravitational-wave physics
Mathematica has become an important tool in general relativity. Packages such as xAct and
components of the Black Hole Perturbation Toolkit manipulate curvature tensors, metric
perturbations, Kerr and Schwarzschild geometries, self-force calculations, and gravitational-wave
equations.
Some modern calculations first determine quantities around black holes numerically to extremely
high precision. Integer-relation methods can then reconstruct exact analytic coefficients
involving constants such as π, logarithms, and zeta values.
In these problems the symbolic expressions can become so large that computer algebra is no
longer merely convenient. It becomes an essential part of doing the physics.
7. Feynman-diagram calculations that would be impractical by hand
The Mathematica package FeynCalc has long been used for symbolic quantum-field-theory
calculations. It can manipulate Dirac matrices, Lorentz tensors, loop integrals, color algebra,
and enormous expressions generated by Feynman diagrams.
A researcher can preserve exact symbolic structure while performing thousands or millions of
algebraic manipulations that would be extraordinarily error-prone by hand.
8. Modeling gravitational-wave detectors
Mathematica has also been used in the engineering behind gravitational-wave observatories.
Symbolic models of the complicated multi-stage pendulum suspensions used in LIGO were developed
with Mathematica and converted into forms suitable for further control-system simulation.
This is an important reminder that Mathematica's contribution is not limited to pure
mathematics. Its combination of symbolic equations, numerical simulation, and programmable
notebooks makes it useful for real physical systems as well.
9. Rule 30: complexity arising from an almost trivial rule
Stephen Wolfram's experiments with cellular automata revealed that extraordinarily simple rules
can produce behavior that looks effectively random.
Rule 30 is one of the most striking examples. Starting from a single black cell and repeatedly
applying a tiny local rule produces an intricate, partly random-looking structure.
This discovery predates Mathematica itself, but the desire to explore large spaces of simple
computational systems helped motivate Wolfram to build the software that eventually became
Mathematica.
10. Rule 110: a tiny rule capable of universal computation
Another elementary cellular automaton, Rule 110, produces persistent structures that collide and
interact in complicated ways.
Wolfram conjectured that it could support universal computation, and Matthew Cook later proved
that it could. Thus one of only 256 elementary binary nearest-neighbor cellular automata is
powerful enough, with suitable initial conditions, to emulate arbitrary computation.
It is difficult to imagine anyone guessing such a property merely by staring at the rule table.
Large-scale computational experimentation made the hidden complexity visible.
11. The Borwein integrals: when a computer pattern suddenly fails
Experimental mathematics can also expose the danger of trusting patterns too readily.
A famous sequence of sinc-function integrals produces exactly π/2 again and again as more
factors are added. The natural conjecture is that the pattern continues forever.
Then, after several cases, the next integral differs from π/2 by an extraordinarily tiny
amount—only around 10−11.
With ordinary numerical precision the answer may still appear to be exactly π/2. Only careful
high-precision computation reveals that the beautiful pattern has finally broken.
What Mathematica really changed
The most important contribution of Mathematica is therefore not simply that it can evaluate a
difficult integral or solve an equation faster than a human.
It allows mathematical work to proceed in a continuous cycle:
formulate symbolically → calculate exactly → experiment numerically →
visualize → notice a pattern → conjecture → verify → prove
In traditional mathematics, computation often came after the creative insight. In experimental
mathematics, computation can participate in producing the insight itself.
That may ultimately be one of Mathematica's most important accomplishments: helping turn the
computer from a machine that merely carries out mathematics into a tool with which mathematicians
and physicists can discover mathematics.
