Bender Bending Rodriguez's Page - Mathematical Curiosities
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 Fry's defrosting Fry was frozen on January 1, 2000 at 0:00 AM. Then, a 1000 years countdown started. The problem is that there exist different years (tropical, sidereal, julian, gregorian...), each one with a particular length. It is natural to consider the "average gregorian year", which has 365.2425 days and it is used in our calendars (that are called precisely "gregorian calendars"). So, 1000 years have 365242.5 days, and then, Fry's defrosting will occur on December 31, 2999 at 12:00 PM (taking into account leap years). Effectively, Fry is woken up on December 31, 2999 and, although the exact hour is not explicitly indicated, it seems that it occurs at midday.

 What day is it today? Bender mentions in episode "1ACV01 - Space Pilot 3000" that the Museum is free on Tuesdays. Precisely, December 31, 2999 is Tuesday. This can be easily proved taking into account that there are 365242 days between January 1, 2000 (Saturday) and December 31, 2999. It can be checked too using this Script to calculate the day of the week or directly looking at 2999 Calendar.

 Rather a "dull" number Bender is the son #1729 (see episode "2ACV04 - XMas Story"). Moreover, the Nimbus ship (that appeared for the first time in episode "1ACV04 - Love's Labours Lost In Space") also has the number 1729. And there also exists the "Universe 1729", as we can see in episode "4ACV15 - The Farnsworth Parabox". The number 1729 is called the Hardy-Ramanujan number, and it is the smallest nontrivial taxicab number, i.e., the smallest natural number representable in two ways as a sum of two cubes: 1729 = Ta(2) = 13 + 123 = 93 + 103. The nth taxicab number Ta(n) is the smallest natural number representable in n ways as a sum of positive cubes. The number derives its name from the following story G. H. Hardy told about Ramanujan. "Once, in the taxi from London, Hardy noticed its number, 1729. He must have thought about it a little because he entered the room where Ramanujan lay in bed and, with scarcely a hello, blurted out his disappointment with it. It was, he declared, 'rather a dull number,' adding that he hoped that wasn't a bad omen. 'No, Hardy,' said Ramanujan, 'it is a very interesting number. It is the smallest number expressible as the sum of two [positive] cubes in two different ways.'" Nowadays, the Taxicab numbers are given by: Ta(1) = 2 Ta(2) = 1729 Ta(3) = 87539319 Ta(4) = 6963472309248 Ta(5) = 48988659276962496 The sixth taxicab number is strongly believed but not yet proven to be 24153319581254312065344. Visit http://euler.free.fr/taxicab.htm to keep you informed. Ta(3) appears on the taxicab that takes Fry in "Bender's Big Score". A Taxicab number in a taxicab! Ta (3) = 87539319 = 1673 + 4363 = 2283 + 4233 = 2553 + 4143 Where will Ta(4) appear? More information in MathWorld.

 Billionary interest The interest that the bank gave to Fry in episode "1ACV06 - A Fishful of Dollars" are, more or less, correct: Initial money = 93 cents; 2'25% of annual interest rate, during 1000 years. Final money = 0'93 * (1'0225)1000 since the amount of money is multiplied by 1.0225 each year. We obtain 4283508449 dollars and 71 cents. This result is rather approximated to 4300 million dollars.

 Geometric buildings In episode "1ACV09 - The Hell Is Other Robots" there appear two buildings with particular geometric forms: "Madison Cube Garden" (a remodelation of "Madison Square Garden") and Hotel "Trump Trapezoid", with cubic and trapezoidal forms respectively. The first one appears in many other episodes.

 Math of Wonton Burrito Meals Professor H. Farnsworth uses a diagram to explain a lesson of his assignature in Mars University (Mathematics of Quantum Neutrino Fields). According to DVD commentaries (episode "1ACV11 - Mars University"), this diagram was provided by David Schiminovich, a physicist at Cal-Tech, and it is a parody of a real particle physics diagram (Witten's Dog references Schrödinger's Cat). Farnsworth concludes that, by process of elimination, the electron must taste like grapeade. The original diagram is from Edward Witten, an important physicist-mathematician that is currently Professor of Mathematical Physics at the Institute for Advanced Study at Princeton, New Jersey (USA). His main works involve string theory and supersimmetry. Precisely, the dog of this diagram is made by strings, representing elementary particle trajectories. More information, here or here.

 Related serial numbers The serial numbers of Bender and Flexo (see "2ACV06 - The Lesser of Two Evils"), can be expressed as the sum of two cubes: Flexo: 3370318 = 1193 + 1193 Bender: 2716057 = 9523 + (-951)3 Moreover, this decomposition is unique.

 The nearest fuel When Fry and Amy are driving down in Mercury (episode "2ACV07 - Put Your Head on my Shoulder"), they run out of fuel just in a place where the nearest (and the only) fuel is at 4750 miles. So, this fuel is exactly at the opposite point on the planet (antipodes), since 4750 miles is half of the length of Mercury's equator. Therefore, whatever the direction we choose, there is always 4750 miles from us to the fuel (drawing a geodesic on Mercury's surface), because Mercury is almost a perfect sphere. *Note: Moreover, "Hg" is the chemical symbol for Mercury.

 A million question In episode "2ACV07 - Put Your Head on my Shoulder", there appears two misterious books labelled "P" and "NP" respectively. They seem to be a recopilation of problems of P and NP classes resp. A problem is assigned to the P (Polynomial time) class if there exists at least one algorithm to solve that problem, such that the number of steps of the algorithm is bounded by a polynomial in n, where n is the length of the input. A problem is assigned to the NP (Nondeterministic Polynomial time) class if it is solvable in polynomial time by a nondeterministic Turing machine. A P-problem is always also NP. If a problem is known to be NP, and a solution to the problem is somehow known, then demonstrating the correctness of the solution can always be reduced to a single P (polynomial time) verification. If P and NP are not equivalent, then the solution of NP-problems requires (in the worst case) an exhaustive search. It is not yet proved that NP = P. If you know the answer, congratulations! You have won a million dollars. There is some progress, and it is proved that "P = NP" is equivalent to "give an algorithm of polynomial time for solving the famous Minesweeper game". Or we could solve the problem taking a look into that pair of books and checking if they are equal. Taking into account their width, the answer seems to be affirmative... More information: MathWorld, Wikipedia.

 Discreet and discrete Bender's computer dating service is advertised as being "Discreet and Discrete" (see episode "2ACV07 - Put Your Head on my Shoulder"), the first meaning exercising self-restraint, and the second a form of mathematics based in logic and computability, which as a robot, Bender is programmed to abide. Discrete mathematics, also called finite mathematics, is the study of mathematical structures that are fundamentally discrete, in the sense of not supporting or requiring the notion of continuity. Most, if not all, of the objects studied in finite mathematics are countable sets, such as integers, finite graphs, and formal languages. (Extracted from Wikipedia.)

 The smallest infinite The cinema of the episode "2ACV08 - Raging Bender" is called "Loew's ℵ0-Plex". It also appears in the episode "3ACV15 - I Dated a Robot". ℵ0 ("Alef null") is a symbol used to denote the cardinality (i.e., the number of elements) of the set of all natural numbers N={0, 1, 2, 3, ...}. So, it is a countably infinite. ℵ1 is the cardinality of the power set of N, i.e. the set whose elements are all the subsets of natural numbers. So, ℵ1 = 2^ℵ0. Moreover, ℵ1 is the cardinality of real numbers, which is a non-countably infinite. The Continuum Hipotesis precisely affirms this fact, and so, there are not any infinite between ℵ0 and ℵ1. In general, ℵn is the cardinality of the power set of the power set... (n times) of N. Recursively: ℵn = 2^ℵn-1 (this is the generalized continuum hypothesis). Since the suffix "-Plex" indicates the number of screens of a cinema (for example, a 12-Plex cinema has 12 screens), we can conclude that the Loew cinema has infinite screens, but countably.

 Schrödinger's Cat The club that professor Farnsworth designs when he was young in the episode "2ACV10 - A Clone of my Own" is called "Schrödinger's Kit Kat Club". Moreover, in the early s.XIX, there exists in London an exclusive club of rich and powerful men named "The Kit-Cat Club" (thanks to Jason for the information). Schrödinger's cat is a seemingly paradoxical thought experiment devised by Erwin Schrödinger that attempts to illustrate the incompleteness of an early interpretation of quantum mechanics when going from subatomic to macroscopic systems. Let us suppose that we have a system formed by a closed and opaque box with a cat inside, a bottle with poisonous gas, a radioactive atom with a 50% of possibility to decay, and a device such that, if the atom decays, then it breaks the bottle and the cat dies. Since all the system depends on the final state of a single atom whose behaviour is given by quantum mechanics, both cat and atom are part of a system subdued to the laws of quantum mechanics. Following the Copenhagen interpretation, while the box is closed, the system simultaneously exists in a superposition of the states "decayed atom/dead cat" and "undecayed atom/living cat", and that only when the box is opened and an observation performed does the wave function collapse into one of the two states.

 Astronomical number In episode "2ACV15 - The Problem With Popplers" it is shown that Fishy Joe's has served over 3.8 x 1010 Popplers. Coincidence or not, this is the mean distance between Earth and the Moon, measured in centimeters. So, if we suppose that the width of a Poppler is 2 cm. and we put all of them in single file, this would get you to the moon and back, which might actually make a better promotional slogan anyway: "We've sold enough popplers to go to the moon and back!" (thanks to Brian). The final number of served Popplers (mentioned by Kif) is 198 billion, i.e. 1.98 x 1011, five times the quantity before. Moreover, the mean distance between Earth and the Moon increases 3.8 cm each year (that's a coincidence!).

 The Binary Beast In episode "2ACV18 - The Honking", there appears the cipher 1010011010 reflected in a mirror. This cipher is 666 in binary numeral system: 1010011010 = 1 * 29 + 0 * 28 + 1 * 27 + 0 * 26 + 0 * 25 + 1 * 24 + 1 * 23 + 0 * 22 + 1 * 21 + 0 * 20 = 512 + 128 + 16 + 8 + 2 = 666. Besides, in the Comic #13 "The Bender You Say" there appears again the number 666 in binary, in the license plate of Robot Devil's car, but this time in the form 0110-0110-0110, that corresponds to 6-6-6 in decimal numeral system.

 Winner... in a quantum finish The final of the horse race of episode "3ACV04 - The Luck of the Fryrish" is so tight that there are only a few particles between winner and second. Then, professor Farnsworth protests claiming that they have changed the outcome by measuring it. He is right, since the Heisenberg Uncertainty Principle (given in 1927) states that when measuring position and velocity (or momentum) of a single elementary particle, increasing the accuracy of the measurement of one quantity increases the uncertainty of the simultaneous measurement of the other quantity. So, if they have also measured the velocity of the horses crossing the finish line, they have changed the position measurement. The fact that measuring quantities of a particle can affect the particle itself is also given in the "wave-particle duality": if we set up an experiment that requires light photons or electrons to behave as particles, they behave as particles; but if we set up an experiment that requires them to act as waves, they act as waves, appearing some wave properties that are incompatible with particles, such as diffraction. More information in Wikipedia.

 The Center and the Edge of the Universe The "Center of the Universe" has always been a very important concept for mankind because of its philosophical transcendence. Nevertheless, the cosmological model inferred from General Relativity describes an isotropic Universe in which there are not privileged points, and it has been confirmed by astronomical observations. That is, the Universe looks practically the same in all directions (at great scale), independently of the point we are observing from. Because of this property, if I set up an experiment to find the Center of the Universe, I will obtain that such center is at me! For example, if I want to find the center of mass of the Universe, I have to make a list of all galaxies, with their masses and distances, and surprisingly, the center of mass will be at the Milky Way (the galaxy where I am). But if I repeat the calculation from another galaxy, I will obtain that the center of mass is at this galaxy too. This has two possible interpretations: the Center of the Universe is any point, or such center does not exist. There happens something similar with the Edge of the Universe: it does not exist due to the fact that the Universe is finite but has no limits. This can be easily understood with the help of the famous balloon analogy, in which the Universe is represented by the two-dimensional surface of a balloon (only the surface, regarding points off the surface as not being part of the Universe at all, and so we are considering a universe with only two spatial dimensions). The galaxies are points homogeneously drawn over the surface of the balloon and the observers in those galaxies can only observe in two spatial dimensions. So, although the Universe is finite, it has no edges, there is not any privileged point, and all observes will observe the same. We have to note that the center of the ballon itself is not on the surface and should not be thought of as the Center of the Universe. In fact, it does not correspond to anything physical. And if it is an expanding balloon, we have an interpretation of the expansion of the Universe...

 Non-orientable beer El envase de la "cerveza de Klein" (ver "3ACV12 - La Ruta de Todo Mal") es la versión en ℜ3 de la curiosa "botella de Klein", una superficie no orientable en ℜ4. Esta versión tridimensional en realidad no es una superficie "suave" debido a que se corta a sí misma; en cambio, la verdadera botella de Klein cuadridimensional no se corta a sí misma y por lo tanto sí que es "suave". El hecho de que no sea orientable quiere decir que la cara de dentro y la de fuera son en realidad la misma cara (esto mismo pasa con la famosa "banda de Moëbius" en ℜ3). Como prueba de ello, si le diésemos vueltas a la botella, la cerveza que contiene se derramaría, cosa que no ocurriría si el envase fuese orientable (como por ejemplo una esfera o un toro, que tienen dos caras: la de dentro y la de fuera). Llegados a este punto, podeis pensar: "Bueno, si usamos como envase una botella normal sin tapón, al girarla también se caería la cerveza...". La diferencia es que una "botella normal sin tapón" no es una superficie "suave", ya que tiene bordes. Si le ponemos un tapón para quitar los bordes, entonces es orientable y la cerveza no caería. Otras marcas de cerveza que aparecen son "Olde Fortran" y "St. Pauli's Exclusion Principle Girl". La primera hace referencia al viejo lenguaje de programación Fortran (que significa "Formula Translation"), utilizado en gran parte por matemáticos (más información en Curiosidades Informáticas). La segunda marca de cervezas es una parodia de la existente marca de cerveza "St. Pauli" (lo de "Girl" es porque esta marca de cerveza organiza un concurso anual para elegir a la "Chica St. Pauli"). Es un juego de palabras con el "Principio de Exclusión de Pauli", un conocido principio de Física Cuántica enunciado por Wolfgang Pauli, ganador del Premio Nobel de Física en 1945: dos partículas distintas no pueden ocupar simultáneamente la misma posición cuántica.

Awesome speed
 When Cubert and Dwight order the Planet Express crew to make a fake delivery to a planet at the edge of the universe in episode "3ACV12 - The Route of All Evil", they waste only 7 days to arrive and return. Since the age of the universe is estimated around 14000 million years, we can suppose that "the edge of the universe" is 14000 million light-years away. So, they have travelled at an average speed of 1460×109 times the old light speed (don't forget that this is possible in the year 3000 because the speed of light has been increased in 2208), that is 4.38×1017 km/s. For example, let us calculate the travelling time to some places of the universe:

 Place Estimated distance Estimated travelling time The Moon 380000 km 0.868 picosec (1 picosec = 10-12 sec) Mars [78×106 km , 378×106 km] [0.178 nanosec , 0.863 nanosec] (1 nanosec = 10-9 sec) Alpha Centauri 4.26 light-years 92 microsec (1 microsec = 10-6 sec) Omicron Persei 8 1000 light-years 21.6 milisec (1 milisec = 10-3 sec) The Center of Milky Way 26000 light-years 0.562 sec Andromeda 2.5×106 light-years 54 sec Quasar 3C 273 2.44×109 light-years 14.6 hours

Por otro lado, el número de cuenta de banco que aparece en el spam relacionado con la lotería nacional española en "El Gran Golpe De Bender" es precisamente la antigua velocidad de la luz expresada en m/s, es decir, 299792458, que son aproximadamente 1079252849 (algo más de mil millones) km/h (gracias a Enrique G.).

 Orbital cooling En el episodio "4ACV08 - Crímenes del Sofocón" se acaba con el problema del calentamiento global alejando a la Tierra del Sol, de forma que el año tiene una semana más. Estudiemos esto con más detalle: La Primera Ley de Kepler dice que todas las órbitas son elípticas con el Sol en uno de sus focos. Para la Tierra, esta elipse es prácticamente una circunferencia, con un semieje mayor de 1 UA (1 Unidad Astronómica = 149.6×106 km) y variando su distancia al Sol entre 0.98 UA (cuando la Tierra está en su perihelio) y 1.02 UA (cuando la Tierra está en su afelio) aproximadamente. La Tercera Ley de Kepler dice P2=K*R3, en donde P es el período de la órbita (es decir, la duración del año), R es la longitud del semieje mayor de la órbita, y K es una constante que depende de las unidades empleadas. En el caso de la Tierra, se suelen emplear las unidades de año sidéreo (365.256363 días) y UA, porque así K=1. Por lo tanto, si incrementamos P en una semana, entonces el nuevo semieje mayor sería de 1.0127 UA, es decir, se vería incrementado en casi 2 millones de kilómetros. Así pues, aunque 2 millones de kilómetros son muchos kilómetros, la alteración producida sería prácticamente tres veces menor que la variación que actualmente sufre la distancia Tierra-Sol a lo largo del año. Teniendo en cuenta que prácticamente no hay diferencias entre las temperaturas de los veranos del Hemisferio Norte (que se producen en el afelio) y los del Hemisferio Sur (en el perihelio), es probable que esta alteración de la órbita no causase un efecto notable sobre la temperatura del planeta. No obstante, como en astronomía no se pueden hacer experimentos, esto no se puede comprobar.

1, 2, 3... to the past!
 El código que permite viajar al pasado en "Bender's Big Score" parece a simple vista difícil de memorizar debido a que es una matriz cuadrada de 6 filas y 6 columnas rellena de 0s y 1s, que hacen un total de 36 dígitos binarios. Pero podemos salvar esta aparente dificultad si nos damos cuenta de que esta matriz es simétrica respecto del eje medio vertical y antisimétrica (es decir, simétrica pero transformando los 0s en 1s y viceversa) respecto del eje medio horizontal. Por lo tanto, sólo tendríamos que memorizar el primer "cuadrante", ya que a partir de éste se puede reconstruir fácilmente toda la matriz:

0 0 1
0 1 0
0 1 1

La tarea se puede simplificar todavía más si convertimos estas cifras de binario a decimal, ya que 0 0 1 = 1, 0 1 0 = 2, y 0 1 1 = 3. Así pues, el código del viaje al pasado no es más que una simple cuenta: ¡1, 2, 3!
Por otro lado 100=4, 101=5, y 110=6, con lo que esta matriz también puede interpretarse como los números binarios del 1 al 6 puestos frente a un espejo.

 Primos Marcianos En la pizarra de Farnsworth y Wernstrom, en "La Bestia Con Mil Millones De Espaldas", aparece un número al que denominan "Martian Prime". Este número es en realidad un número primo de Mersenne (nótese la similitud fonética entre "Martian" y "Mersenne" en inglés) con 39 dígitos, descubierto por Edouard Lucas en 1879, y fue el número primo más grande conocido hasta la era de las computadoras en 1950. Los números de Mersenne Mn son aquellos de la forma 2n-1 con n natural, y para que sea primo se requiere que n sea también primo, excepto para n = 2 (pero no todo Mn con n primo es primo). Hasta la fecha (septiembre de 2008), solamente se conocen 46 primos de Mersenne, y el más grande es precisamente el número primo más grande conocido: 243112609-1, con casi 13 millones de cifras. El número que aparece en la pizarra es el primo de Mersenne número 12, y corresponde a 2127-1. Además, este número tiene otra particularidad, y es que 127 también es un primo de Mersenne. En realidad, podemos hacer una sucesión usando la regla Xn = 2Xn-1-1 con X1 el primer primo de Mersenne, es decir 3 = 22-1. Así pues, esta sucesión continuaría de la forma X2 = 7, X3 = 127, y nuestro primo marciano sería X4. Todos estos números son también primos de Mersenne, con lo que X5 también debería ser un primo de Mersenne, pero esto es una conjetura y no está demostrada... Más información en la Wikipedia. Por otro lado, en la pizarra aparece "Goldbach" y un mensaje en Alien-1 que dice "quodlibet": Quodlíbet es una pieza de música que combina diferentes melodías en contrapunto, usualmente temas populares, y a menudo en forma sencilla. Un ejemplo muy conocido se encuentra en el final de las Variaciones Goldberg, Variación Nº 30, de Bach (extraído de la wikipedia). Con "Goldberg" y "Bach" se puede construir "Goldbach", que es el autor de la famosa conjetura que dice que todo número par se puede descomponer como suma de dos primos. Se supone que Farnsworth y Wernstrom han demostrado esta conjetura de forma sencilla mediante un quodlíbet y con la ayuda de los números primos "marcianos"...

 The E-Tunnel El túnel que atraviesa la nave de Planet Express (en la secuencia de opening de Bender's Game) está compuesto por las cifras decimales del número e. Este número irracional, llamado también constante de Neper, es importante porque es el único número real que usado como base de una función exponencial hace que la derivada de ésta (es decir, su pendiente) en cualquier punto coincida con el valor de dicha función en ese punto. Es decir, la derivada de la función f(x) = ex es también ex. El número e es uno de los números más importantes en la matemática, junto con el número π, la unidad imaginaria i y el 0 y el 1. Curiosamente, la identidad de Euler los relaciona (eπi+1=0) de manera asombrosa. Más información en la wikipedia. Se puede consultar el primer millón de decimales del número e en esta página.

 Another million question La hipótesis de Riemann, formulada por primera vez por Bernhard Riemann en 1859, es una conjetura sobre la distribución de los ceros de la función zeta de Riemann. Es uno de los problemas abiertos más importante de las matemáticas contemporáneas (y parece que seguirá abierto en el siglo XXXI, como podemos ver en el "Cómic #11 - The Cure for the Common Clod") y el Clay Mathematics Institute ha ofrecido un premio de un millón de dólares por una demostración (como en el problema P=NP), ya que resolverá muchas incógnitas acerca de la distribución no aleatoria de los números primos. La función zeta de Riemann está definida para todo número complejo z distinto de 1. Tiene algunos ceros llamados "triviales" para z = - 2, z = -4, z = -6, ... La hipótesis de Riemann se centra en los ceros no triviales, y enuncia que "la parte real de todo cero no trivial de la función zeta de Riemann es 1/2". En el plano complejo, la recta Re(z)=1/2 (en donde se supone que se encuentran todos los ceros no triviales) se denomina "recta crítica". Más información en MathWorld.

Irrational numbers
 El Canal de Noticias Raíz de 2, en varios episodios, como por ejemplo "1ACV08 - Un Enorme Montón de Basura" o "2ACV03 - A la Cabeza de las Elecciones".
 La Histórica Raíz de 66 en "3ACV02 - Parásitos Perdidos". "Route" ("ruta" en inglés) se pronuncia muy parecido a "Root" ("raíz" en inglés). Una lata de aceite lubricante π-in-1 en "3ACV11 - Mal del Ordenador Central". La πth Avenue después de la 3rd Avenue, en "3ACV21 - Acciones Futuras". La marca de muebles y complementos del hogar πKea en "4ACV04 - Menos que un Héroe". Concurso "¿Cuál es el último dígito de π?" en el "Cómic #13 - The Bender You Say". El problema es que al ser un número irracional, no tiene último dígito.

Plantilla de Futurama con títulos universitarios de ciencias:

• J. Stewart Burns: Licenciado en Matemáticas por la Universidad de Harvard y Máster en Matemáticas por U.C. Berkeley. Productor y Guionista de Futurama.
• David X. Cohen: Licenciado en Física por la Universidad de Harvard y Máster en Ciencias Computacionales por U.C. Berkeley. Productor Ejecutivo y Guionista de Futurama.
• Ken Keeler: Doctor en Matemática Aplicada por la Universidad de Harvard y Máster en Ingeniería Electrónica. Productor Ejecutivo y Guionista de Futurama.
• Bill Odenkirk: Doctor en Química Inorgánica por la Universidad de Princeton. Guionista de Futurama.
• Jeff Westbrook: Doctor en Ciencias Computacionales por la Universidad de Princeton. Guionista de Futurama.

Más información en Futurama πk.