The Mathematics Tarot Handbook

THEMATHEMATICS

TAROT

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An Anchorite Studios Production

The Mathematics Tarotby Karen West

Unless otherwise noted, all images in this deck are Public Domain or used under a Creative Commons license. Please check the source image and attributions before using these images for your own projects, as the licenses and permissions are subject to change.

Thank you for your support!

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ACKNOWLEDGEMENTS

This deck is dedicated to my madgirl Spark of a sister, without whom it would not exist. Adolina, no matter what time of year it is when you read this...Merry Christmas! This one's for you.

I would also like to thank my parents for their unfaltering support, and for not murdering me when I blathering about the magnificence of the Fibonacci sequence for the five hundred and eleventh time.

Lastly I would like to thank Kyle, for being a precogniscant bastard and telling me that I'd fall in love with math...eventually. The odds were against you, mate, but you were right in the end.

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Contents

: Contents

Page 7: Introduction

Page 13: Majors

Page 41: Minors

Page 43: Cups

Page 53: Coins

Page 63: Wands

Page 73: Swords

Page 83: Courts

Page 101: Spreads

Page 105: Games and Tricks

Page 109: Further Reading

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INTRODUCTION

It is the last day of the semester, and I am fifteen years old.

When a counsellor approaches me and asks me why I'm crying, I despairingly tell her that I'm certain I've failed the semester on account of my math course, and shall have to take the year over.

She takes me to her office, gives me a cup of tea and some tissues, and says, "Now, I'm really not supposed to do this, but we'll make an exception in this case, so that you don't give yourself a heart attack with all this worry." She proceeds to check the exam results on her computer.

I passed that course at 51%.

"I hate math," my fifteen year old self hisses in that office, with all the impotent passion of a child, echoing the sentiments of a million others.

After that summer, I dropped out of public schooling for medical reasons, and spent the next five years trying to complete my education through distance education programs with varying levels of success. It wasn't until I was twenty-one that I finally came close to graduation, with only one barrier standing between me and my future.

Yes, you guessed it.

A math course.

This time, I was mature enough to realize that there was only one way through this: if I could make myself like math, if I could find some redeeming quality or interesting fact about it, then I could convince myself to finish that damn course and get on with my life. I embarked upon a mission to the internet, where I found documentaries, first on particle physics (which was interesting enough) and then later, on mathematics.

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I was utterly astounded by what I found.

I felt like someone had hit me upside the head with a brick and involved me in a vast, absurd conspiracy, intended to convince the world that math was useless and frustrating and uninteresting—when in reality, math is none of those things. The history of mathematics is full of enough weirdness, murder, intrigue, and triumph to inspire ten seasons of soap operas and docudramas. It was staffed with magnificently flawed geniuses who worshipped numbers, or duelled with pistols; men who turned down million-dollar prizes and the women who programmed the first computers; palaces built as love letters to symmetry and mysteries that would remain forever unsolved. Even the rough beast of practical mathematics was itself deep and sanguine and fascinating, if you knew where to look, and knew why the numbers did what they did.

I got angry, then.

When I had taken mathematics in school, many students, including myself, had asked the perennial question: what good was all this rubbish? What was it for? What good would it do us? And the answer, of course, was put forth to us again and again, by different teachers over the years: "Well, you won't use any of this unless you become a rocket scientist, but you have to learn it anyway." Generation upon generation of students, taught that the class they're taking has no purpose beyond receiving an arbitrary slip of paper at the end of the year. No teacher I studied under ever attempted to make mathematics fun, or taught us any of the vibrant history, or even showed us math games; the students I grew up with would have loved conning their friends with those. I blame the system and not the teachers for this—god knows they were harried enough as it was, with overlarge classes and unwieldy schedules. Needless to say, if you're a teacher, you're a hero and a gentleperson, and never forget it.

As I worked my way through that final calculus course, shocking myself with B's and occasionally A's after a lifetime of C-'s and near-failures, I tried to cure the people around me of their hatred for mathematics. I focused more on the women in my life, since the girls in my classes had been especially picked out as superfluous—girls don't become mathematicians, of course! I shared my new-found love of mathematics with my foster sister, who was rapidly working her way through calculus and physics courses of her own, aiming for a position as a naval engineer in the Canadian Forces.

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We spent many evenings with mixed drinks and mixed company, exasperating our friends by swapping obscure mathematics jokes back and forth for hours, and excitedly talking over each other about the most recent book we'd read on prime numbers.

And then one fateful day, after a tiny, ridiculous light-bulb flashed on above my head, I sent my sister a text message that read: "I'm thinking of making a mathematics tarot deck. SAVE ME FROM MYSELF."

Her wildly enthusiastic response (and possibly a handful of death threats if I didn't go through with it) is why I'm here today, writing a handbook for the Mathematics Tarot.

It's an incredibly surreal experience for me, to be honest. A few years ago, if you'd suggested that I would so much as appreciate math, let alone loving it and even getting half-decent grades, I would probably have called the sanatorium. And if you'd told me I would be publishing a math-themed tarot deck, I would have fainted in a pile of petticoats. As I write this, the deck has not yet been released for sale, but the online response to the concept and images has blown me away. A huge thank-you goes to everyone who has liked or commented upon my silly work-in-progress posts about the deck; you all helped keep me sustained and inspired throughout the whole crazy process, and allayed my fears about creating this deck, despite being an amateur enthusiast instead of an authoritative expert.

Whether you're a tarot collector with no particular love for mathematics, or whether you're a curious mathematician, I hope you enjoy this weird little deck, and I sincerely hope that I've done both the maths itself—and the tarot format—sufficient justice. Thank you so much for purchasing it and supporting a self-employed student.

Happy reading!

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The Cards

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The Majors

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The FoolLorenz System

The Math: A series of equations known to have chaotic solutions given certain initial values and conditions. The image is the visualized plot of a Lorenz attractor, arguably the most famous Lorenz solution, and often referred to as the Butterfly. If you type "Lorenz attractor" into the search engine of your choice, you can find many beautiful images of the chaotic forms these equations create, as well as applets which allow you to create your own.

Interpretation: Mania, chaos, eccentricity, transcendence, original impulses, strangeness, insanity.

Image Source: Public Domain.

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The MagicianUncertainty Principle

The Math: Often confused with the Observer Effect of quantum physics, the uncertainty principle states that the more precisely one measures the position of a particle, the less precisely one is able to measure its momentum (and vice versa). One cannot know both the position and momentum of a particle simultaneously. This frustrating bit of probability lead to Einstein's famous remark, loosely translated as "I am convinced that God does not play dice," to which Bohr replied, "Einstein, don't tell God what to do."

Interpretation: Subtlety, spooky action at a distance, skill, cunning, transaction, interference, deceit, elasticity.


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The PopessSine Wave

The Math: Also called a sinusoid, the sine wave is much simpler than the complex waveforms produced by musical instruments, and illustrates a smooth, repetitive oscillation. All waves can be built by sine waves, and are behind many mathematical and physical phenomena: sound, light, radio, aquatic, seismic. Sine waves, in a term, are natural sway.

Interpretation: Fluctuation, purity, change, alternation, increase and decrease, balance.

Image Source: http://commons.wikimedia.org/wiki/File:Sine_wave_amplitude_wavelength.svg

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The EmpressPotential Energy

The Math: The energy of an object or system which is conserved and transferred by position or state; in other words, an object can store energy as a result of its position. The elastic you draw back between your hands to ping your best friend in class holds a great deal of potential energy, as your friend is about to discover. Although potential energy can exist in relation to other fundamental forces, the equation here is for gravitational potential energy; m for mass, g for the gravitational constant, and h for height.

Interpretation: Potential, activity, gravity, earthiness, completeness, love, beauty, simplicity.


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The EmperorMass-Energy Equivalence

The Math: Arguably the most famous and well-known piece of mathematics in history, Einstein's equation states that the mass of an object and its energy are connected by a fixed constant; the faster an object goes, the greater its mass and energy, and vice versa. Einstein put forth the Mass-Energy Equivalence along with four other groundbreaking papers in 1905, while still working as an examiner at a patent office after several years of fruitless attempts to find a teaching job. These four papers granted him a professorship six years later, while his fame slowly and deservedly began to spread.

Interpretation: Solidity, energy, a firm foothold on reality, fixed proportions, ambition, originality, reality.


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The HierophantPythagorean Theorem

The Math: As famous as E=mc2 but not quite as iconic in pop culture, this theorem was discovered algebraically as many as 4000 years ago, and is a fundamental part of Euclidean geometry. It states that for any right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Although most well-known for its use in geometry, this simple equation can be used for any shape or formula in which a square number is used, and can even be used to measure distance, ratings, or personal taste.

Interpretation: Wisdom, orthodoxy, authority, goodness, endurance, strength, teaching, manifestation, peace.

Image Source: http://commons.wikimedia.org/wiki/File:Pitagorasteorema.svg

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The LoversThe Mandelbrot Set

The Math: Loved outside of mathematics for its visual appeal, this set is a dynamical system in which the output of a function is fed back into the equation as its input, resulting in an infinitely detailed boundary with a central void. If you zoom into any area of the set, you will find the same shapes repeated in new and more complex arrangements the deeper you go.

Interpretation: Development, duality, self replication, alchemy, connection, choice, harmony, divergence.


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The ChariotInflation

The Math: Cosmic inflation is the widely-accepted theory of rapid exponential expansion put forth to explain the mysteriously flat, homogeneous appearance of our universe, when according to Big Bang cosmology it should be appear curved and heterogeneous.

Interpretation: Victory, authority under authority, obedience, traditionalism, swiftness, fame.

Image Source: http://commons.wikimedia.org/wiki/File:Horizonte_inflacionario.svg

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JusticeEntanglement

The Math: Information appears to be able to pass instantaneously between two particles or molecules, irrespective of distance; therefore measuring one particle of the pair has a correlating effect on the state of the other particle, even if it is absurdly far away from the first. This seeming paradox lead Einstein to call quantum mechanics incomplete, but Bell later proved it to be inarguable.

Interpretation: Equilibrium, impartiality, fundamental laws, adjustment, agreements, treaties, suspension of action.


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The HermitMinkowski Spacetime

The Math: A setting in which the three regular dimensions of space are combined with one dimension of time, forming a spacetime manifold. The diagram for this is fairly widely recognized as a "light cone" or "time cone" in which the bottom of the hourglass shape represents the converging past, and the top represents the ever-widening future.

Interpretation: Time, introspection, analysis, movement, illumination, light.


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The Wheel of FortunePi

The Math: Simply put, the ratio of a circle's circumference to its diameter, represented by the Greek letter of the same name. Pi is a favourite number of many, having a great deal of fun properties: it is in irrational number, as its decimal representation never ends or repeats, and is also transcendental, as you cannot make a whole number from it no matter what you add, subtract, multiply, or divide into it. Memorizing and reciting the digits of pi is a popular hobby among mathematics enthusiasts and record-breakers.

Interpretation: Eternal change, revolutions, unexpected events, good fortune, abundance.


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StrengthCurie's Law

The Math: The law that the magnetization of a substance is inversely proportional to its temperature. While gravity works over a longer distance, magnetism is by far the stronger force, as one can easily test by "levitating" a paper clip from a table with a simple fridge magnet.

Interpretation: Force, strength, controlled power, resolve, tenacity, energy, action, courage, overcoming weakness.


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The Hanged ManThe Higgs Boson

The Math: A massive particle, once considered hypothetical, predicted by the Standard Model to explain why fundamental particles have mass. Scientists have been searching for the Higgs since the 1960's, but given the difficulty of detecting unstable particles in the accelerator before they decay, it was not until 2012 that a particle which matches the expected signature for the Higgs was discovered. While not conclusively the eponymous Higgs, its behaviour is consistent with scientific predictions about the boson and its place in the Standard Model. Its nickname, the God Particle, is widely disliked by scientists—though its namer, Leon Lederman, once joked that he would rather have called it the Goddamn Particle, given the level of frustration and expense it has caused the scientific community.

Interpretation: Trials, devotion, defeat, sacrifice, re-evaluation, perspective, suspension.


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DeathBinary

The Math: Also called a base-two numerical system, binary notation has only two symbols: zero and one. Developed as early as the 5th or 2nd century for linguistic use, similarly to Morse Code, as well as being a vital component of I Ching hexagrams, binary is now used in modern computing and in the realms of logic and ciphers.

Interpretation: Duality, choice, yes/no, on/off, life/death, change, transformation, development.


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TemperanceEuler's Formula

The Math: Widely felt to be the most beautiful formula in mathematics, Euler's equation links five fundamental constants of mathematics (zero, one, pi, i, and e), as well as using the three basic arithmetic operations once each (addition, multiplication, and exponentiation). It is generally written as eiπ + 1 = 0.

Interpretation: Unity, ideas, accommodation, good combinations, calculation, realization.

Image Source: http://commons.wikimedia.org/wiki/File:Euler%27s_formula%28vi%29.svg

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The DevilMaxwell's Demon

The Math: The Demon is a character in a thought experiment regarding the Second Law of Thermodynamics—which asserts that in an isolated system, entropy never decreases—and shows how to hypothetically violate that law. The Demon controls a door between two insulated containers, and carefully opens the door to allow only faster, hotter particles to enter one container, while isolating slower, cooler particles in the other, thereby decreasing entropy.

Interpretation: Disorder, temptation, obsession, the taboo, forbidden knowledge, discontent, secret plans, ambition, unscrupulous behaviour, power.


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The TowerFermat Conjecture

The Math: For three hundred and fifty-eight years, mathematicians around the world struggled to prove Fermat's Last Theorem, which Fermat himself scribbled a reference to in the margins of a book in 1637, saying that the proof itself was too large to fit. It was finally solved by Andrew Wiles in 1995, after six years of secret work. It states that given an equation an + bn = cn, no integer value of n greater than 2 can satisfy the equation, hence the Pythagorean equation is the limit (a2 + b2 = c2).

Interpretation: Frustration, purgatory, redemptive suffering, quarrel, escape from imprisonment, truth.


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The StarThe Riemann Hypothesis

The Math: The mathematician David Hilbert famously said, "If I were to awaken after having slept for a thousand years, my first question would be: Has the Riemann hypothesis been proven?" The hypothesis is a conjecture regarding the location of nontrivial zeros of the zeta function, but in practical terms, implies answers to the age-old conundrum about the distribution of prime numbers. It occupies a healthy place on Hilbert's list of unsolved problems, and is one of only three problems remaining that are considered both unresolved and potentially possible to solve.

Interpretation: Faith, hope, immortality, bright promise, possibilities, insight, clarity of vision, disappointment if ill dignified.


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The MoonDivide By Zero

The Math: Anyone who has taken high school or college algebra knows what happens when you accidentally (or intentionally) divide by zero on your calculator: you get an error message. In mathematics, dividing by zero can range from mildly frustrating, in the case of a small algebraic error, to catastrophic, in the case of the USS Yorktown incident, in which a database field error caused the ship's propulsion system to fail. Dividing by zero has become a meme in popular culture on the internet, and is often used to humorous effect on photographs of sinkholes and bell-mouth spillways. Dividing by zero can also produce the 2=1 fallacy, given some simple algebraic tweaking.

Interpretation: Error, deception, danger, illusion, calumny, bewilderment, madness, falsehood.

Image Source: http://commons.wikimedia.org/wiki/File:Hyperbola_one_over_x.svg

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The SunThe Golden mean

The Math: Famous in both mathematics and art, the golden mean, or golden ratio, states that two quantities are in the golden mean if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one—or, in simpler terms, a+b is to a as a is to b. The diagram is of the Fibonacci spiral, a related piece of maths; if you take a Fibonacci number and divide it by the previous Fibonacci number, you get closer and closer to the golden mean. The Fibonacci sequence and the golden mean appear everywhere in maths, art, architecture, nature, music, finances, and design.

Interpretation: Happiness, glory, beauty, truth, triumph, pleasure, recovery, pleasant surprise.


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JudgmentIsometries

The Math: Also known as translations, isometries are movements of an object on a plane without distorting the object in question. If you've taken high school math, you'll remember moving shapes and parabolas about on a graph: rotations, slides, reflections, and glides.

Interpretation: Transformation, change of position, renewal, definite steps, decisions.


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The WorldTheory of Everything

The Math: While the theory itself is simply an umbrella term for various theories that attempt to unite disparate areas of physics, the concept has high aims. Currently, although both work perfectly well while separate, the theory of general relativity and the theory of quantum mechanics don't quite mesh, and the sticky issue is gravity. Many theories have been put forward, some more promising than others, but so far none have come close to answering the penultimate question of science: is there a theory that links the entirety of physical phenomena? Can science ever explain the whole universe?

Interpretation: The universe, large and lofty issues, synthesis, crystallization, victory, the end of the matter, celebration, independence.


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The Empty Set

The Math: A seemingly useless piece of mathematics, the empty set is exactly what it sounds like. It is not the same as nothing, but is rather a set with nothing in it; an empty bag still exists in and of itself. This concept is surprisingly useful in a few niche areas of mathematics, but it's more commonly known for its philosophical curiosities. D.J. Darling describes the empty set as "the set of all triangles with four sides, the set of all numbers that are bigger than nine but smaller than eight, and the set of all opening moves in chess that involve a king."

Interpretation: apparent absurdity, paradox, void, potential, curiosity.


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Möbius

The Math: Most of us have made a moebius strip by taking a long strip of paper, twisting one end, and taping it to the other end, and if we haven't, we've probably seen the diagram in which an ant walks around the strip forever without coming to an edge. It's one of those shapes that seems impossible in the real world at first glance, and later is found to be deceptively simple. A similar object is the klein bottle, a beautiful and somewhat miraculous glass object with one continuous surface which contains itself.

Interpretation: What goes around comes around, infinity, everything is connected, ordinary miracles.

Image Source: http://commons.wikimedia.org/wiki/File:Moebius_strip.svg

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The Mathematician

Interpretation: This card may be used in the deck proper as a regular card, or removed for use as a significator, representing the reader or querent. In general readings, this card represents skill, craft, vocation, the things you are good at, and the similarities between your career and your hobbies. If the Mathematician comes up while you're procrastinating your homework... well. You know what to do.


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The Minors

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Ace of CupsStrong Interaction

The Math: Also called the strong nuclear force, this fundamental interaction is much stronger than electromagnetism at an atomic level. Carried by gluons, it binds protons and neutrons together to form the nucleus of an atom, and at a lower level, binds quarks together to form protons, neutrons, and hadrons.

Interpretation: Attraction, peaceful interaction, productiveness, beauty.

Image Source: http://commons.wikimedia.org/wiki/File:Pn_scatter_pi0.png

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Two of CupsEuler-Euclid Theorem

The Math: Mersenne numbers are any positive integer that is one less than a power of two. Mersenne primes, by extrapolation, are Mersenne numbers that are prime, of which there are currently only 47 known. A perfect number is any positive integer that is equal to the sum of its factors (for instance, 6 is a perfect number, as you can add 1+2+3 to reach it). The connection between Mersenne primes and perfect numbers is mysterious and fascinating: if M is a Mersenne Prime, [M(M+1)] ÷ 2 will be an even perfect number every time. Discoveries of new Mersenne primes immediately result in new perfect numbers.

Interpretation: Harmony, unity, combination, marriage, pleasure.


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Three of CupsSierpinski Triangle

The Math: Also called the Sierpinski gasket or Sierpinski sieve, this aesthetic fractal is constructed in a deceptively simple way. Draw a triangle, then draw the largest possible reversed triangle in the centre. Continue filling up the largest available space with upside-down triangles until the size of your pen nib limits you. Like the Mandelbrot set, this is a self-similar fractal, meaning that you can magnify infinitely and find the same patterns. While this fractal can be drawn with other shapes, such as squares, the triangle is the most attractive and clean result; it can also be formed in three dimensions, making a beautiful lacy pyramid.

Interpretation: Abundance, plenty, success, kindness, new perspective, liberality.


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Four of CupsThe Center of a Triangle

The Math: If I give you a piece of paper with a circle or a square on it, and ask you to find the centre, I imagine you'd point it out without hesitation—but if I gave you a triangle and asked you to find the centre, you might hit a stumbling block. In fact, according to the Encyclopedia of Triangle Centers, there are over five thousand answers to this fiendish question, a number which is growing all the time. Taking one triangle, you can pick from handfuls of different points to measure from, and each will give you a centre ever so slightly different from all the others you tested.

Interpretation: Acquisition via struggle, stationary happiness, pleasure but not without discomfort, pursuit.


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Five of CupsContinuum Hypothesis

The Math: Some areas of mathematics seem specifically designed to send people mad; cardinal numbers are one of those things. Cantor demonstrated that there are infinitely many levels of infinity, and insisted that for every set of numbers, of which there are infinitely many, there was a cardinal number to go with it. The Continuum Hypothesis, rumoured to have driven Cantor to severe depression, asks whether there is another cardinal number in between that of the integers and the real numbers. It was the first of Hilbert's list of 23 unsolved problems, and was later determined to be fundamentally unprovable.

Interpretation: Disappointment, an end, engagements broken off, trouble from unexpected sources, unexpected loss.

Image Source: http://commons.wikimedia.org/wiki/File:ContinuumHypothesis.svg

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Six of CupsNewton's Second Law

The Math: While the First Law simply states that force is required to accelerate an object, the Second Law answers the question about how much force is required. The necessary force is directly proportional to the object's mass; the larger the mass, the more force is required.

Interpretation: Smooth increase, beginnings, patience, advancement, steadfastness.


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Seven of CupsPrinciple of Explosion

The Math: This principle has nothing to do with gunpowder and everything to do with logic. If you translate the logic symbols on the face of this card, it reads "if something is claimed to be both true and untrue, you can derive any conclusion," also known as "from a contradiction, anything follows," similar to the way you can prove that two equals one using division by zero.

Interpretation: Lies, promises unfulfilled, error, vanity, selfish dissipation, deception.


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Eight of CupsUnstable Equilibrium

The Math: While stable equilibrium is represented by a ball in a ditch, which will return to its resting position despite small perturbations, unstable equilibrium can be upset by the slightest of disturbances. Imagine the difference between a marble resting at the bottom of a funnel, and a pencil balancing on its tip—like a house of cards which crumples when you breathe on it, the system is primed for collapse.

Interpretation: Instability, decline, temporary success, things which do not last.


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Nine of CupsMean Value Theorem

The Math: Simply put, given an arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints. In other words, at some point the slope of a curve must equal its average slope. You can imagine this using the analogy of a car travelling down a highway: if its average speed throughout the trip was 100km/h, at some point the car must have been travelling at exactly 100km/h, even if the rest of the trip was taken at slower or faster speeds. Though it seems simple enough, this theorem had widespread and mind-shattering consequences throughout mathematics, especially in the realm of real numbers and continuous functions, as well as being essential to proving the fundamental theorem of calculus.

Interpretation: Success, pleasure, not being satisfied with small ideas, fulfilled wishes, lovability.

Image Source: http://commons.wikimedia.org/wiki/File:Mvt2.svg

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Ten of CupsFundamental Theorem of Arithmetic

The Math: Also known as the unique-prime-factorization theorem, this theorem states that any integer greater than one can be written as a unique product of prime numbers. It also says two important things about number theory: first, that any positive whole number can be broken down into prime factors, and secondly, that this can only happen in one way. Notably, these rules do not apply to rational numbers (fractions). These simple points are the underlying structure of the whole of mathematics.

Interpretation: Permanent and lasting success, happiness, quiet victory, heavenly inspiration.

Image Source: Public Domain

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Ace of CoinsGravity

The Math: Though we generally think gravity to be the strongest force in the universe, it is in fact the weakest of the four fundamental forces! Without it, however, there would be no universe, no planets, and no us. Gravitation is the phenomenon by which objects attract, with a force proportional to their mass. It gives weight to everything within its field, which is why you would weigh less or more on a planet with a different mass. Gravity is less-well understood than laymen assume, with a long list of anomalies attached to it, and is also the main bone of contention in the attempt to merge general relativity with quantum mechanics.

Interpretation: earthiness, reality, things which are hard to grasp, Goldilocks zone.

Image Source: http://en.wikipedia.org/wiki/File:NewtonsLawOfUniversalGravitation.svg

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Two of CoinsHailstone numbers

The Math: The main element of the Collatz Conjecture, hailstone numbers are fun to play with. Start by choosing any whole number and apply these rules: if it is even, halve it, and if it is odd, triple it and add one. Continue to apply this rule until you go down to 1 . Therein lies the magic, because it appears as though you can use these two rules to get to 1 regardless of which number you start with—hence the name "hailstone" numbers, for they always fall to the ground no matter how high you toss them. The Collatz Conjecture asks whether this is true for every whole number, which is just as hairy as asking how many digits there are in pi; Paul Erdos allegedly said of this conjecture, "Mathematics is not yet ripe for such problems."

Interpretation: Change, alternate gain and loss, wandering, journeys which end well, waffling.


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Three of CoinsPerfect Magic Cubes

The Math: Magic cubes in themselves are curious enough; they take the basic idea from magic squares, which function a little like sudoku, and go up one dimension to a cube. Perfect magic cubes take it further than that: not only the columns, rows, pillars and regular diagonals add up to the same number, but so do the cross section diagonals. Interestingly, perfect cubes with two to four orders aren't possible, but cubes of five orders and up have been discovered as early as the 1800's and as late as the 2000's. The image on the card is a cube with an order of five; if you were to cut out the pieces and slot them behind each other, you would create the cube itself.

Interpretation: Cleverness, creation, building, constructive force, work, seeking after impossibilities.


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Four of CoinsPerelman's Theorem

The Math: Also called the Poincare Conjecture after its originator, it remained a conjecture for nearly a century, and made the list of seven Millennium Problems. While unsolved, it was arguably the most important question in topology, the study of continuous deformations of objects without tearing them (source of the old joke about how topologists are mathematicians who don't know the difference between a donut and a coffee cup). In topology, if you can make a loop on an object and tighten it to a point continuously, then the object is considered a sphere, which is the only simply connected surface known in topology. The conjecture asked whether the same was true if you went up a dimension to three-dimensional spheres. Between 2002 and 2003, Grigori Perelman solved the conjecture, proving that the same is indeed true for 3-spheres, but famously turned down both the $1,000,000 prize and the Fields Medal.

Interpretation: Success, rank, gain of money or influence, completion, endings.

Image Source: http://en.wikipedia.org/wiki/File:P1S2all.jpg

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Five of CoinsUltrafinitism

The Math: This mathematical philosophy, which comes in a variety of flavors, denies the existence of the infinite set of natural numbers, on the basis that it is impossible to complete a list and therefore it should not be attempted. Many ultrafinitists also deny extremely large numbers, since they are considered so unwieldy as to be useless. An extreme offshoot of constructivism, some varieties of ultrafinitism go so far as to state that mathematical objects do not exist unless constructed from natural numbers in a finite number of steps. Of course, the problem is, where do you draw the line? If 2 raised to the power of 100 is too large, then what about 2 raised to the power of 99? 98? 54?

Interpretation: Obstinance, lack of imagination, trouble regarding material objects, poverty, anxiety, loss.


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Six of CoinsPerfect Numbers

The Math: A whole number is considered perfect if all its factors (not including the number itself) add up to reach it. 28, for instance, is a perfect number: 1 + 2 + 4 + 7 + 14 = 28. Mathematically inclined cultures have often attached spiritual significance to these numbers, of which there are only 47 known to date. As discussed in the Two of Cups, perfect numbers and Mersenne primes are intimately connected, so there are as many known perfect numbers as there are known Mersenne primes. Currently all known perfect numbers are even, and while no odd perfect numbers have yet been discovered, the possibility that they are simply exceedingly rare still exists.

Interpretation: Material success, prosperity, good business deals, influence, nobility, justice.


58

Seven of CoinsHilbert's 4th Problem

The Math: This problem is simply the search for new non-Euclidean geometries (or geometries that stand outside of the non-Euclidean area of geometry) through geodesics, which is the study of which sort of lines are the shortest distance between two points. The answer isn't always as easy as it sounds—try making a straight line on a torus. While something of a solution to the general concept was given by Georg Hamel, the original statement has been considered too vague to answer fully.

Interpretation: Disappointment, little gain for much labour, loss of promising future, isolated gain.


59

Eight of CoinsThe frivolous theorem of arithmetic

The Math: Like Discordianism, which is a religion disguised as a joke disguised as a religion, this theorem is serious mathematics disguised as wit disguised as a theorem. It goes like this: "Almost all natural numbers are very, very large." No matter how large the numbers we discover happen to be, they are shadowed into miniscule absurdity by all the numbers beyond them. There will always be bigger numbers. This theorem, as silly and trivially self-evident as it seems, had important implications for the realm of data compression.

Interpretation: Skill, cunning, hoarding, a lack of enterprise, great things disguised as trivial ones, focus on the small at the expense of the large.


60

Nine of Coins1 – 2 + 3 – 4 + . . .

The Math: A series comprising all positive integers using alternate signs, this sequence is notable for visiting every integer—even zero and the negatives numbers!—exactly once. It does not tend towards a finite limit; it is, in fact, an infinite set.

Interpretation: Complete material gain, goods, riches, increase, treasure, inheritance.

Image Source: http://en.wikipedia.org/wiki/File:Pm1234_Ground.png

61

Ten of CoinsSporadic Groups

The Math: Some consider the classification of finite simple groups the most incredible achievement in mathematics, and some consider that mathematicians have too much time on their hands. A finite group is a group which has a finite number of elements, and this classification—a worldwide group effort comprising 500 papers by over a hundred mathematicians—states that every finite simple group belongs to one of four categories. It also states that there are only 18 countably infinite families and 26 sporadic groups in total, from which all other finite groups are built, which is a spectacularly remarkable achievement. The largest sporadic group has been fittingly dubbed the Monster, and has in its claws a mind-boggling 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 elements.

Interpretation: Completion of gain, ultimate fortune, pinnacle of success, cleverness, heaviness, loftiness.

Image Source: http://en.wikipedia.org/wiki/File:Finitesubgroups.svg

62

Ace of WandsElectromagnetism

The Math: Once upon a time, electricity and magnetism were viewed as two separate forces, and both were used in exclusion of the other by both sideshow entertainers and serious scientists. Then, one day, the current of a battery was observed to interfere with the needle of a compass, and the rest is history. Far from simply the spark behind electric motors, electromagnetism is present in our everyday lives in surprising ways—in fact, nearly all physical phenomena except those under the umbrella of gravity are caused by electromagnetism. From binding electrons into their orbits to chemical processes, electromagnetism is the fundamental force that makes everything tick.

Interpretation: Energy, the spark of creation, creativity, natural force, harmonious combination.

Image Source: http://commons.wikimedia.org/wiki/File:Feynmann_Diagram_Coulomb.svg

63

Two of WandsVenn Diagram

The Math: Named after the man who invented them in the late 1800's, Venn diagrams are used to show the relations between two sets. They aren't relegated to the realm of mathematics, but are heavily used in linguistics and other social sciences. Although the most common Venn diagram has two spheres that overlap in the centre, Venn diagrams can easily compare five or more sets, given a complex arrangement of shapes invented by Venn and Edwards.

Interpretation: Authority, boldness, courage, resolution, ambition, power over the unknown.


64

Three of WandsTupper's Self-Referential theorem

The Math: While it has little practical purpose, this formula is an interesting exercise in graphing. If one inserts the inequality formula and an absurdly large given value for n into a graphing program, the graph will be an exact, if somewhat pixelated replica of the original formula. Altering the formula slightly to give it more height simultaneously improves the resolution, to the point where the pixelation disappears and a clean font emerges on the graph.

Interpretation: Proudness, self-assertion, success after struggle, nobility, inward focus.


65

Four of Wandsadding up 1 to 100

The Math: A charming story is told of this bit of mathematics. Carl Friedrich Gauss, as a wee small boy, was in math class one day when his teacher asked the class to count to 100 using all the previous numbers: 1 + 2 + 3 + 4, and so on to 100. Expecting a restful, quiet class while his students added away, the teacher was baffled when little Gauss stood up within moments, the answer in his hand. You can do this clever trick yourself with any number by replacing n in the equation with whatever number you wish to add up to—in this case, to 100. (The answer, by the way, is 5050.)

Interpretation: Cleverness, reasoning, wit, rest after labour, completion of a project, helpful shortcuts.


66

Five of WandsLeibniz-Newton Calculus Controversy

The Math: Begun in the late 1600's and only moderately wound up by the time Leibniz died in near anonymity in 1716 (his grave remained unmarked for fifty years), the Controversy began when these two scientific giants apparently independently discovered calculus. Debate raged fiercely for years over who had been first, and whether Leibniz had simply come up with a different method of notation, while Newton invented the core idea. In an age without digital records and timestamps, the only evidence was the word of each man. At first, it appeared as though the matter would end peacefully, but an anonymous tract suggesting that Newton had stolen Leibniz’s ideas created an inferno. While neither man wished for fame, both wished to retain their dignity and integrity before the scientific world, and were unable to back down without losing their standing. Today, it is generally accepted that each invented the calculus independently.

Interpretation: Quarrelling, fighting, strife, indignity, boldness, prodigality, genius, desire.


67

Six of WandsSchrödinger equation

The Math: As Newton's Laws are to classical mechanics, Schrodinger's Equation is to quantum mechanics. It describes how the quantum state of a system changes over time, and its implications created breakthroughs throughout the world of earth 20th century physics. Overall, it isn't terribly difficult to understand: the equation can be read as total energy equals kinetic energy plus potential energy. It has additional implications in the realm of particle measurement: when measured, a particle's properties are randomly taken from a "smear" of probabilities, and while the equation cannot predict exact results (this is, in fact, fundamentally impossible), it can still predict what the probability distributions will be. Hence, it also has a great deal to do with the incredible particle-wave behaviour which is most famously visible in the double-slit experiment.

Interpretation: Pleasure gained by labour, success after strife, gain, carefulness, victory, important matters.


68

Seven of WandsNewton's Third Law

The Math: While this equation can be quoted by every schoolchild as "every action has an equal and opposite reaction," this statement can be misleading. When dealing with some activities, such as bouncing a ball, it makes sense, but what about hitting a nail with a hammer? Does the nail effect the hammer with the same force the hammer exerts on the nail? Indeed it does! If you're adventurous, you can test this out by stepping out of a boat without taking much care; as you push out of the boat, the boat pushes back, leaving you somewhat damp and bedraggled between canoe and beach. Similarly, the reason we don't collapse to the floor under the effects of gravity is because the ground responds with the same upward force.

Interpretation: Influence, equality, wrangling with equal victors, the courage to meet difficulties.

Image Source: http://commons.wikimedia.org/wiki/File:BookNormal.svg

69

Eight of WandsSpeed of Light

The Math: Measured in the universal metres per second, this number is so large as to be difficult to understand; it may be easier to envision the imperial version of 186,282 miles per second. On second thought, perhaps not! However you measure it, light travels at mind-bogglingly extraordinary speeds, such that it takes only a little over eight minutes for light from the sun to reach us, despite the intervening 149,597,870 miles. While light can travel through glass and air easily, its speed is slowed ever so slightly by passing through them. A strange thing to think about while laying on your lawn after sunset is that you're looking back in time; the light from stars takes thousands, millions, even billions of light-years to reach our eyes. This is partly how scientists are able to study events which happened in the universe's deep past. To give you an idea of the sheer brain-breaking size of the universe, consider that in a single Earth year, light can travel nine thousand, four hundred and sixty one billion miles. Multiply that by four to get to the closest star, Proxima Centauri.

Interpretation: swiftness, speed, travel, hasty communications, sudden force, generosity.


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Nine of WandsOctonion

The Math: Octonions require a little back-story, as they are wonderfully quirky. First off, to get a complex number you must add imaginary numbers to real numbers. Complex numbers form the end of what we consider the evolution of numbers; they're the pinnacle of Everest, and hold the world of numbers on their metaphorical shoulders. 6 + 2i, is, for example, a complex number. Now, evolution doesn't have to stop. It isn't necessarily logical, but you can keep going—which resulted in quaternions. These strange numbers are the next evolution of complex numbers, and contain four terms instead of two: 6 + i2 + j4 + k5, for example, the squares of which equal -1. This discovery made some areas of mathematics more complex, as it is now not always true that AxB = BxA. The octonion is an even greater leap, and consists of eight terms instead of two or four, and the squares of all the terms still, magnificently, equal -1. The octonion, really, despite its usefulness in Lie groups and quantum logic, is the crazy uncle no one lets out of the attic. He's just a little too much for everyone to handle.

Interpretation: questions, intractability, unexpected strength, steady yet quiet force, unexpected success, eccentricity.

Image Source: http://commons.wikimedia.org/wiki/File:FanoMnemonic.PNG

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Ten of WandsBlack hole

The Math: Ask a gathering of children about these, and at least one of them will probably tell you, with gruesome glee, about how you'd be spaghettified if you floated near one. (The real theory is only slightly less horrifying.) In any case, black holes are widely agreed to be strewn liberally around the universe, and many scientists have reason to believe that a massive one exists at the centre of every galaxy. In layman's terms, a black hole is an area of space with a gravitational field strong enough to devour even light, caused by the collapse of a star. For a few decades, black holes were merely a mathematical curiosity, but recently they have become the subject of major investigation, especially in the realm of entropy, which the card's equation is about. The Hawking-Bekenstein equation confirms that black hole entropy is proportional to the area of its event horizon—the point of no return—divided by the Planck area. This was a necessary piece of mathematics, given that if black holes had no entropy, interaction with them would violate the second law of thermodynamics.

Interpretation: overbearing force and energy, opposition too strong to be controlled, selfish ends, failure, defeat, futility.


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Ace of SwordsWeak Nuclear force

The Math: Also called weak interaction, this fundamental force is the impetus behind radioactive decay. Called "weak" because of its exceedingly short range of influence, it is nevertheless a powerful force, which effects fermions and is responsible for hydrogen fusion in stars. It is unique among the forces for being able to change the properties of one quark—known as its "flavour"—into that of another. Although radioactive decay is often exceedingly dangerous, and something most of us would rather not play with, radiocarbon dating of ancient artifacts would be impossible without it, as would the existence of the universe itself.

Interpretation: Repellance, decay, weakness, short-sightedness, asymmetry, the centre cannot hold, danger.


73

Two of SwordsStrange equality

The Math: In a seeming contradiction that has defied logic (and the patience of math students) for untold years, the repeating decimal 0.999 is actually the number 1. Now, you might say, "Well, of course it is! You round up!" But in fact, the mystery remains, because you don't. By some quirk of mathematics, the two numbers are actually equal, and there exist a huge list of proofs (of varying rigour and believability) of this odd truth. Some summarize that the proof exists simply to drive future generations of students to drink, but so far, no one has been able to ask the great mathematician in the sky whether that was the intention.

Interpretation: Truth and untruth, debate, repeating affronts, quarrels settled yet tensions remain, uneasy truce.


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Three of SwordsSchrödinger's cat

The Math: What appears at first glance to be an extraordinarily complex attempt at animal torment is actually a thought experiment based on entanglement and quantum mechanics. The thought experiment, which is now famous, places a cat in a box with a radioactive source and a vial of poison; if an internal Geiger counter detects radiation, it smashes the vial, thus killing the cat. Or does it? Similar to the age-old question of whether a tree makes a sound if it falls when no one is around to hear it, the experiment suggests that if the box remains unopened, the cat remains in a permanent liminal state—both living and dead simultaneously. When the box is opened, the wave function collapses, and the experiment is decided one way or the other. Until the flap is lifted, all the possibilities are smeared out in probability distribution, and only an observer will force them to settle into a frustratingly definite state, leaving little information about the in-between distribution.

Interpretation: Interruption, disruption, quarrelling, discord, strife, mischief-making, sadistic pleasures.


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Four of SwordsTrachtenberg arithmetic

The Math: The Trachtenberg systems are forms of rapid mental calculation, but not for any old numbers—these brilliant systems work best for multiplying extremely large numbers with great ease. Each number from 3 to 12, excepting 10, has an easily memorized set of calculations attached to it, which make multiplying them by unwieldy, six-figure (or more!) numbers quick and simple, by means of addition, carrying, and simple rules. Trachtenberg himself was a Russian Jew during the age of Nazism, a mathematician and an engineer and, obviously, a genius, who developed these systems to entertain his brain during his seven-year imprisonment in a concentration camp. He later founded the Mathematical Institute of Zurich, one of the most famous schools of mathematics in modern history.

Interpretation: Peace after war, recovery, good from sorrow, change for the better, redemption.


76

Five of SwordsSeven Bridges of königsberg

The Math: Look at the image on the card. Now imagine that each of those white arches are bridges in a city, and you have been charged with the task of moving through the city while only crossing each bridge once. You can't go over a bridge twice, or backtrack, or fly. Can you do it? The answer—although I don't blame you if you try a few times regardless—is no. It is a fundamentally unsolvable problem. (Unless, of course, you take the advice of my fey young cousin who, upon being faced with this problem, said with absolute dignity and assurance, "Just build another stupid bridge.")

Interpretation: Defeat, contest lost, thanklessness, trouble, loss, anxiety, insomnia, pity.

Image Source: http://commons.wikimedia.org/wiki/File:7_bridges.svg

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Six of SwordsMersenne Primes

The Math: Mersenne primes are named after their discoverer, a French monk. A Mersenne number is any positive number which is one less than a power of two (thirty-one, for example). A Mersenne prime, of course, is simply a Mersenne number which is also prime, which is any natural number greater than one which has no positive divisors other than 1 and itself—thirteen, for example, is prime. Many of the Mersenne primes have been discovered by the mighty efforts of the Great Internet Mersenne Prime Search (affectionately abbreviated GIMPS), which has been run on exponentially more powerful computers since its release in 1996. Not only has the project discovered thirteen of the Mersenne primes, but eleven of those thirteen were also the largest discovered prime number at the time, breaking records on many occasions.

Interpretation: Good labour, work, journeys, success after effort, victory after trouble, patience, beauty.


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Seven of SwordsString theory

The Math: String theory is one of the most controversial areas of modern physics. Some feel it to be the leading contender for the Theory of Everything which ties together general relativity and quantum mechanics, while others feel it to be a lost cause, and many have staked their career and reputation on it. The basic concept behind string theory is this: at the very bottom of reality, deeper and tinier than anything we know, are not particles, but oscillating lines (or strings). It assumes many extra "curled up" dimensions in order to work. Although it is indeed among the leading contenders for a so-called final theory, it has its troubles. The major criticism lies in the inability to test string theory experimentally, given the unbelievably miniscule proportions of the strings themselves, and it is unlikely that we will ever be able to build a particle accelerator that is able to see the strings, even if they do exist. String theory also has the curse of the cornucopia: there are too many solutions and dimensions, instead of not enough. String theory is extraordinarily compelling, but may in the end be too impractical to answer our fundamental questions.

Interpretation: Partial success, unreliability, unwieldy plans, ambition, love of abundance, yielding before victory.


79

Eight of Swordszeno's paradox

The Math: The first recorded instance of reductio ad absurdum, this paradox pits Achilles against the humble Tortoise in a footrace. The Tortoise reasonably asks for a head start, since of course he is slower than the mighty Achilles, and Achilles, being a upright gentleman, courteously allows this—to his cost. For once Achilles reaches the point at which the Tortoise started, the Tortoise has already moved on another percentage, and once Achilles reaches that point, the Tortoise is already slightly more ahead...and so on. Poor Achilles must run a gamut of infinite points, each decreasing in length as he proceeds, evermore chasing a Tortoise. More a mathematical problem dealing with progressively smaller fractions than a workable philosophical conundrum, Zeno's Paradox is based on the ancient idea that motion is merely an illusion, and that everything is one.

Interpretation: Narrowness, futility, restriction, too much attention to detail, a prison.


80

Nine of SwordsMillennium Prize problems

The Math: Based on Hilbert's list of twenty-three unsolved problems, the Millennium Prize problems are a list of seven fiendish problems that remain unsolved to this day—and, notably, have the potential to be solved—with the notable exception of the Poincare Conjecture (also called the Perelman Theorem after the man who solved it). If you come up with the solution to any of these problems, you will be awarded a whopping $1,000,000 by the Clay Mathematics Institute. These are the most difficult, wicked issues in mathematics, the worst of the worst and the greatest of the great, each of which would have groundbreaking implications in other realms of science if solved. Don't wait around; these beasts might be solved tomorrow, or they might take another hundred years.

Interpretation: Despair, burden, pitilessness, labour, craft, patience, faithfulness.


81

Ten of SwordsUncountable sets

The Math: The natural numbers are all the practical ones we use in daily life: 0, 1, 2, and so forth on to infinity. The real numbers, on the other hand, comprise every single decimal expansion and whole number, on to infinity. While the natural numbers are all whole, the real numbers contain all the numbers in between the natural numbers, as well as the natural numbers themselves. Remember the frivolous theorem of arithmetic, which states that most natural numbers are very, very large? Well, the real numbers are beyond the scale of human comprehension—so large an infinity dwells innocently between 0 and 1 that it dwarfs the natural numbers into absurdity. Don't think about it too hard, or your brain might melt.

Interpretation: Ruin, defeat, failure, repetition, insanity, obsessive behaviour, despair.

Image Source: http://en.wikipedia.org/wiki/File:Latex_real_numbers.svg

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The Courts

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Princess of CupsZero Simplex

The Math: A 0-simplex is a single point with no dimensions.

Interpretation: Simplicity, seeds, embryonic situations, destiny, potential.


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Prince of CupsOne Simplex

The Math: A 1-simplex is a line segment bounded by two points, with a single dimension.

Interpretation: Equality, beginnings and endings, boundaries, lack of perspective.


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Queen of CupsTwo Simplex

The Math: A 2-simplex is a triangle bounded by three line segments and three points, with two dimensions.

Interpretation: Sharpness, acuity, moving up in the world, learning, intelligence.


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King of CupsThree Simplex

The Math: A 3-simplex is a tetrahedron composed of four triangles, six line segments, and four points, which has three dimensions.

Interpretation: Reinforcement, constancy, steadfastness, preparation, the pinnacle, success.


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Princess of CoinsCircle

The Math: Euclid's definition of the circle, which has survived the intervening years of poetic mathematicians, is that a circle will appear when you mark a spot on the ground and ask ten people to stand around it at a distance of one metre. We've all dealt with circular math in our younger years of schooling: the circumference, the diameter, the arc, the radius, the chord, and of course, the ubiquitous pi, the ratio of a circle's circumference to its diameter.

Interpretation: Connectedness, smoothness, perfection, beauty, yet instability.


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Prince of CoinsSquare

The Math: A square is generally defined as a regular quadrilateral with four equal sides and four equal angles. Squares are a favourite of schoolchildren, as they require less advanced mathematics to measure than other shapes, given their equal proportions. The value for the diagonals of a square is also known as Pythagoras' constant, and was the very first irrational number to be proven.

Interpretation: Stability, groundedness, lack of imagination, stubbornness, permanence.


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Queen of CoinsPolygon

The Math: Within the jurisdiction of this Queen's rule are triangles, quadrilaterals, and pentagons, as they are all two-dimensional shapes with straight lines that meet at corners. In fact, you can draw just about any shape with straight lines, and classify it based on the number of sides it has. The black sheep of the family are the digon, a polygon with two sides and two corners; that is, a line—and the apeirogon, which is composed of an infinite number of line segments and has an equally unlimited number of sides.

Interpretation: Abundance, common knowledge, service, vitality, changeability.


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King of Coinspolyhedron

The Math: These three-dimensional shapes, while vastly different, have one thing in common: they are all a collection of two-dimensional faces with straight edges meeting at corners. The simplest polyhedron is the pyramid, and the forms move upward through the cube, the icosahedron pictured on the card face, all the way up to absurdly complex shapes such as the great disnub dirhombidodecahedron (known less awkwardly as Skilling's Figure) which has 360 total edges.

Interpretation: fertility, platonic ideals, complexity, playfulness, enthusiasm.


92

Princess of Wandsaddition

The Math: Addition is one of the very first things we learn to do as babies. I have one block in front of me, and papa puts another block in front of me—how many do I have? Two! Addition is simply counting, one of the most intuitive human skills, and something so basic that many animals have been observed to do it. Addition is interesting because it is commutative: order doesn't matter.

Interpretation: Increase, stable growth, growing things, children, production.


93

Prince of WandsSubtraction

The Math: Subtraction is slightly more mysterious. All is well and good at first. I have three blocks and I take one away, leaving me with two blocks. But what if I have three blocks and take away five? Suddenly, the blocks in my hands turn into slime and blow raspberries at me. Of course, in mathematics, we would say the answer is -2, or negative two, but this is difficult to elucidate with physical objects. Even stranger, if you minus a negative number (5 minus -10, for example) you actually add it (meaning the answer is actually 15). The first users of subtraction using negative numbers either vilified them as wrong and absurd (in the case of the Greeks) or used them to financial advantage (for calculating debt, in the Arab, Indian, and Chinese world).

Interpretation: Severance, decrease, anticlimax, low tide phase, waning.


94

Queen of WandsMultiplication

The Math: Multiplication is like addition's big sister. If I have three bags, each of which has three blocks in it, how many blocks do I have? I could add 3+3+3, but it's much easier to multiple 3x3, both of which give me the same answer: 9. Basic multiplication is commutive, but the more complex the equation, especially while dealing with complex numbers, the order does indeed matter. One of the most famous artifacts in mathematics, the Ishango bone, which is dated to approximately 18,000 BC, suggests a knowledge of multiplication among peoples we once assumed to have been far too primitive for mathematical understanding.

Interpretation: ascension, exaggeration, strength, climbing, height, lofty ideals.


95

King of WandsDivision

The Math: Division asks the inverse question to multiplication: how many times does this number fit into this other number? Division immediately leads to the concept of rational numbers, or fractions, as not all numbers fit easily into one another. Division is interesting in that it can be performed with a wide variety of methods, from pushing stones about on a board, to the use of an abacus or slide rule, to logarithmic tables. Division is generally the most disliked of the four basic methods of arithmetic among schoolchildren, given how it is closely connected to fractions, as well as the multitude of gradually more complex rules of cross-cancelling and synthetic division which appear in higher maths.

Interpretation: Reduction, tallies, erosion, reservation, leavings, leftovers.


96

Princess of Swordsintegers

The Math: The integers are all the natural numbers (0, 1, 2, 3, etc) combined with their cousins once removed, the negative numbers, or everything below zero. They are a subset of real numbers, and are the numerical basis of all the mathematics you've ever done. There are no fractions or square roots within their ranks; they are fundamentally simple.

Interpretation: Comprehensiveness, all your ducks in a row, abundance, wide choices.

Image Source: http://commons.wikimedia.org/wiki/File:Integers-line.svg

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Prince of Swordsimaginary numbers

The Math: The name is slightly misleading; these numbers are neither "all the numbers you can imagine" nor "numbers which do not exist." They are, in fact, numbers whose square is less than or equal to zero, something which generally doesn't happen in good mathematics, and the whole set is a little bit of a redheaded stepchild. They have the mysterious quality that if you square the imaginary symbol i, you get -1 as your answer, which allows you to write imaginary numbers as real numbers. However, despite their strangeness, without imaginary numbers, mathematics would not have passed very far beyond the seventeenth century.

Interpretation: Creativity, imagination, cleverness, eccentricity, mysteries, tricks.


98

Queen of Swordsrational numbers

The Math: These are, simply put, the fractions—or, as she would be called if I renamed her, the Queen of Quotients. All integers are children of fractions, since they can be written as, for example, 2/1, or two over one. Of course, because you cannot divide by zero, the bottom number (or denominator) of a rational number can never be zero.

Interpretation: Simplification, equality, balance, details, lightness, caution.


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King of Swordscomplex numbers

The Math: A marriage between imaginary numbers and real numbers, you can imagine the complex numbers as seeds who refused to stay on the one-dimensional number line and bloomed into a two-dimensional flower. Used today in a variety of fields from quantum physics to engineering, fractals to fluid dynamics, the complex numbers have been variously loathed and sanctified throughout the centuries.

Interpretation: Blooming, expansion, marriage, agreements, alchemy, new perspective.

Image Source: http://en.wikipedia.org/wiki/File:Complex_number_illustration.svg

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Spreads

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Dimensions spread:One Dimension - where you startTwo Dimensions - where you go from hereThree Dimensions - what you will do when you get there

Fundamental Forces spread:Electromagnetism – what inspires youWeak Force – what pulls you apartStrong Force – what holds you togetherGravity – what grounds you

Euler's identity spread:e – the core question (the base of natural logarithms)i – deeper insight (imaginary unit of complex numbers)pi – how to solve it (analytical mathematics)1 – what you bring to the situation (the multiplicative identity)0 – the outcome (the additive identity)

Isometries spread:Rotation – what surrounds youTranslation – where you are goingReflection – how you see yourselfGlide – how to move forward

Entanglement spread:Draw two cards. You may only look at one of them; when you are finished, replace both cards into the deck.Position – where you are nowMomentum – where you are going

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T.O.E. Spread:General Relativity – the established situationQuantum Mechanics – the new element, the mystery, the challengeFinal Theory – how to bridge the gap or solve the problem

Venn Diagram Spreads:a) Situation A Situation B How they overlap

b) Person A Person B How they get along/are similar

c) Desire A Desire B How to compromise

d) Situation/project Querent Their interaction

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Games & Tricks

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Nim:Invented in the seventeenth century, Nim is a great game to play when you're bored on the beach with a friend. Find yourself a pile of stones, counters, M&M's, or any other collection of small objects (between 15 and 40 is good). Each player takes a turn to remove either one, two, or three counters from the pile. The person to take the last counter wins.

Magic Squares:If you're tired of Sudoku and need a new challenge, try to come up with as many 4x4 magic squares as you can – there are over eight hundred in total. Each row and column must add up to the same number, as well as the diagonals. If even that fails to challenge you, try 5x5's – there are over two hundred million of those.

The 1089 Puzzle:Write down any three digit number that does not read the same backwards and forwards (616, for example, is not allowed). I shall pick 876. Now, reverse the digits: 678. Now, subtract the reversed number from the original: 876 – 678, which equals 198. Reverse this one: 891. Now add the two new numbers together: 198 + 891. The answer is 1089 – and if you don't cheat and use the illegal numbers, you will get 1089 every time. Why, I hear you ask in disbelief? It simply has to do with an interesting quirk in regarding number columns, in which a number in the form abc will always produce the same leftover number in this format, no matter what those numbers are (excepting the palindrome numbers).

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The 11 Trick:We all know how to multiply something by ten; you just add a zero to the end. But how about multiplying by eleven? Surprisingly, it's almost as easy. Say you want to multiply 52 by 11. Take the 5 and the 2 and add them together to get 7. Then sandwich it between the 5 and the 2 like so: 572. And there you have your answer! If the number you add ends up being a two digit number, it's slightly different; say you want to multiply 85 and 11. 8 + 5 is 13. Put the 3 in the middle to get 835, then add the 1 to the 8 to get 935, which is your answer.

The 4 & 5 Tricks:These are brilliant tricks, the sort of thing which makes you wonder how you didn't think of them before. To easily multiply a number by four without relying on your times-tables, simply double it, as if you're multiplying by two, then double it again. 4 x 10 can be done as 20 + 20, which is 40. Similarly, if you want to multiply by 5, just divide the number in half and then multiply it by ten . 5 x 4 can be done as 2 x 10, which is 20. You can do a similar thing if multiplying by 12. Let's try 30 x 12 – first, multiply 30 by 10, which gives you 300, then 30 by 2, which gives you 60, and after you add those two together, your answer is 360.

The Fibonacci Trick:If you need to get a rough estimate of conversion between miles and kilometres, you can use the famous Fibonacci sequence. Yes, it really is everywhere. Three miles is approximately equal to five kilometres, five miles is about eight kilometres

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The Mathematics Tarot Handbook

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