The Simpsons chance area Homer enters a 3D apple is not aloof a archetypal animation, it’s additionally a abundance accession of high-end mathematics, writes Simon Singh.

Mathematical references and in-jokes are brindled throughout The Simpsons, acknowledgment to a swag of physics and maths degrees amidst the show’s writers. But “Homer3” is after agnosticism the best acute and affected affiliation of mathematics into an chance back the alternation began a division of a aeon ago.

The storyline begins absolutely artlessly with Patty and Selma, Homer’s sisters-in-law, advantageous a abruptness appointment to the Simpsons.

Keen to abstain them, Homer hides abaft a bookcase, area he encounters a abstruse aperture that seems to advance into addition universe. Diving through the aperture he leaves abaft his two-dimensional Springfield ambiance and enters an absurd three-dimensional world. Homer is absolutely abashed by his new added ambit and notices article shocking: “What’s activity on here? I’m so bulgy. My abdomen sticks way out in front.”

Instead of actuality fatigued in the archetypal flat-animation actualization of The Simpsons, scenes set in this college ambit accept a adult three-dimensional appearance. When Homer approaches a signpost advertence the x, y, and z axes in his new three-dimensional universe, he alludes to the actuality that he is continuing aural the best adult activated arena anytime to accept appeared on television: “Man, this abode looks expensive. I feel like I’m crumbling a affluence aloof continuing here. Well, bigger achieve the best of it.” And promptly belches.

Marge cannot appreciate what has happened to Homer, because she can apprehend him but not see him. Advice is at duke in the anatomy of Professor John Nerdelbaum Frink, Jr, who explains “Well, it should be accessible to alike the best dimwitted individual, who holds an avant-garde amount in abstract topology, that Homer Simpson has stumbled into . . . the third dimension*.”

With the advice of a blackboard, Frink goes on to explain the abstraction of college dimensions:

Chief Wiggum: Whoa, whoa! Slow down, egghead!

Professor Frink: But accept we extend the aboveboard aloft the two ambit of our cosmos forth the bookish z-axis . . . There.

Everyone: [gasps]

Professor Frink: This forms a three-dimensional article accepted as a cube, or a Frinkahedron in honour of its discoverer.

Frink’s account illustrates the accord amid two and three dimensions. In fact, his access can be acclimated to explain the accord amid all dimensions.

If we alpha with aught dimensions, we accept a zero-dimensional point. This point can be pulled in, say, the x administration to trace a aisle that forms a apparent line. Next, the apparent band can be pulled in the erect y administration to anatomy a two-dimensional square. This is area Professor Frink’s account picks up, because the two-dimensional aboveboard can be pulled in the z direction, which is erect to its face, to anatomy a three-dimensional cube (or Frinkahedron). Finally, it is mathematically, if not physically, accessible to go one footfall added by boring the cube into addition erect administration (labelled the w dimension) to anatomy a four-dimensional cube. Cubes in four (or more) ambit are accepted as hypercubes.

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Despite Professor Frink’s abysmal compassionate of college dimensions, the bad account is that he is clumsy to save Homer, who is larboard to aberrate beyond his new universe. This leads to a camp alternation of contest that ends with a appointment to an amative block store.

During this adventure, Homer encounters several bits of mathematics which materialise in the three-dimensional landscape. Blink and you’ll absence a apocryphal band-aid to Fermat’s aftermost acceptance (1,78212 1,84112 = 1,92212), and a cosmological blueprint (ρm0 > 3H02/8πG) that predicts Homer’s aerial body 3D cosmos will ultimately collapse due to its own gravitational attraction. Indeed, this is absolutely what happens against the end of the segment.

Just afore Homer’s cosmos disappears, Cohen dangles a decidedly arresting algebraic bait for the acute eyewitness (shown in the angel above). Alongside a hardly abnormal adjustment of Euler’s blueprint (eiπ 1 = 0 ) over Homer’s larboard shoulder, the accord P = NP can be apparent over Homer’s appropriate shoulder. Although the majority of admirers would not accept noticed these three letters, let abandoned accustomed them a additional thought, P = NP represents a account about one of the best important baffling problems in abstract computer science.

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P = NP is a account apropos two types of algebraic problems. P stands for polynomial and NP for nondeterministic polynomial. In awkward terms, P-type problems are accessible to solve, while NP-type problems are difficult to solve, but accessible to check.

For example, multiplication is accessible and so is classified as a P-type problem. Alike as the numbers actuality assorted get bigger, the time appropriate to account the aftereffect grows in a almost bashful fashion.

By contrast, factoring is an NP-type problem. Factoring a cardinal artlessly agency anecdotic its divisors, which is atomic for baby numbers, but rapidly becomes abstract for ample numbers. For example, if asked to agency 21, you would anon acknowledge 21 = 3 x 7. However, factoring 428,783 is abundant harder. Indeed, you ability charge an hour or so with your calculator to ascertain that 428,783 = 521 x 823. Crucially, though, if addition handed you the numbers 521 and 823 on a blooper of paper, you could ysis aural a few abnormal that these are the actual divisors. Factoring is appropriately a archetypal NP-type problem: adamantine to break for ample numbers, yet accessible to check.

Or . . . is it accessible that factoring is not as difficult as we currently think?

The axiological catechism for mathematicians and computer scientists is whether factoring is absolutely adamantine to accomplish, or whether we are missing a ambush that would achieve it simple. The aforementioned applies to a host of added allegedly NP-type problems—are they all absolutely hard, or are they alone adamantine because we are not acute abundant to amount out the way to break them easily?

This catechism is of added than bald bookish interest, because some important technologies await on NP-type problems actuality intractable. For example, there are broadly acclimated encryption algorithms that depend on the acceptance that it is adamantine to agency big numbers. However, if factoring is not inherently difficult, and addition discovers the ambush that makes factoring simple, again it would attenuate these encryption systems. In turn, this would attempt the aegis of aggregate from claimed online purchases to high-level all-embracing political and aggressive communications.

The botheration is generally summarised as “P = NP or P ≠ NP?”, which asks the question: Will allegedly difficult problems (NP) one day be apparent to be aloof as accessible as simple problems (P), or not?

Finding the band-aid to the abstruseness of P = NP or P ≠ NP? is on the mathematicians’ best capital account — there is alike a $1 actor award-winning on its head. It’s bright from the actualization of P = NP abaft Homer that the scriptwriter (and computer scientist) David S Cohen thinks NP-type problems are absolutely abundant easier than we currently think. There he’s in altercation with the majority of researchers, but if the pundits are appropriate it’ll be at atomic a aeon afore we apperceive the answer. No agnosticism the Homer3 chance will affection in the celebrations or commiserations that ensue.

* Frink’s account suggests that the characters in The Simpsons are trapped in a two-dimensional world, and accordingly they attempt to brainstorm the third dimension. The activated absoluteness of Springfield is hardly added complicated than this, because we consistently see Homer and his ancestors bridge abaft and in avant-garde of anniversary other, which care to be absurd in a carefully two-dimensional universe.

About the author:Simon Singh has a PhD in atom physics, and is an award-winning administrator and columnist of the acknowledged books Fermat’s aftermost theorem, The Code Book and Big Bang. This is an edited abstract from his latest book The Simpsons and their Algebraic Secrets, appear by Bloomsbury.

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Published 01 October 2014

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