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25 Cards in this Set
- Front
- Back
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David Hilbert |
-From Germany during the 19th century -He made "Hilbert's Hotel" which describes different levels of infinity |
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1-1 Correspondance |
-When 2 sets have the same cardinality,same size, and can be matched up with each other Ex: Set 1= 2, 4, 6, 8 Set 2= 1, 3, 5, 7 |
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Finite, countably infinite, and uncountably infinite |
Finite= can be put into a 1-1 correspondence with the numbers 1, . . . n for a fixed # n countably infinite: cardinality of the natural numbers uncountably infinite: cardinality strictly larger than the natural numbers |
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Cantor's Diagnolization Argument |
-Georg Cantor's proof that if a list of real numbers and a list of natural numbers are made to match, you can always create a number that you missed. Ex: .123 .456 .567 .243 |
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Power Set |
The set of all subsets Example: (123) (1)(2)(3)(12)(13)(23)(123)0 |
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infitesimals |
-an immeasurably small number; not 0, but small than any real number and has infinitely small quantities - related to infinity because they are both seen in fractals |
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Coastline Paradox |
-First looked at by Lewis Richardson and it is the idea that the measurement of a coastline will keep getting bigger as long as a smaller way of measuring it is being used because a coast line is so jagged the smaller ruler you use the more detail you include in the measurement |
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Fractals |
-a shape that exhibits self similarity, can be defined as reclusive steps -self similar and infinitely detailed and if zoomed in it still looks the same Example: a plant, food such as broccoli, a mountain side |
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Fibonacci Sequence |
1, 1, 2, 3, 5, 8, 13. . . -get the next number be adding the previous two -invented by Leonardo Fibonacci -related to the golden ratio b/c 1/2, 2/3, 3/5, etc. gets closer and closer to the exact number of the golden ratio |
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Symmetry |
-a motion or operation that leaves an object unchanged -three types of symmetry are translation, reflection and rotation -an n-gon has n- symmetries. For example, a pentagon has 5 symmetries |
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Spherical Geometry |
-A type of non-euclidean geometry discovered by MC Escher, he used it in his art -The geometry of a sphere and it connect points on a globe rather than on a flat piece of paper |
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Vanishing Point |
-Used to show perspective in picture -A point where paralle lines converge 3 types: Nadir-- point is below the picture, eyes are drawn down Zenith-- point is above the picture, eyes are drawn upward Point on the horizon-- the point is on the horizon line and it draws the eyes to the center of the picture |
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Impossible Figures |
-A 2-D image that looks 3-D but cannot actually exist in a 3-D world Example: Escher piece of art "waterfall" uses the penrose triangle |
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Oragami |
-originated in Japan during 17th century, it is an act of paper folding to create a sculpture. no cuts or tape are used. |
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Akira Yoshizawa |
-From Japan during the 19th Century -he created over 50,000 origami sculptures during this lifetime |
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The types of origami |
Plain/pure: folding 1 sheet Modular: folds multiple sheets the same way and then puts together to create a bigger, more complex piece Action: the model has some movement Wet-folding: the paper is dampened to create more realistic shapes Tesselations: makes tessellation's out of folded paper Mathematical/Technical: allows for more complexity, computer programs can make patterns |
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Charles Bouton |
-Founder of Nim from the USA during 19th Century -developed a complete theory and winning strategy for the game; which is to use binary digits to remain in a safe position |
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Piet Hein |
-founder of 2-D version of Nim, TacTix. -From Denmark during the 20th century |
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Sprague-Grundy Theorem |
-States that every impartial game under normal play is equivalent to a nimber -proved independently by Sprague and Grundy after studying Nim during the 20th century |
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Combinatorial Game Theory |
-It studies strategies and math of perfect information, no chance games. -the games can be impartial or partisan -it uses mathematics and computer science -began when Bouton solved Nim in 1902 |
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Double cross, Long chain theorem |
Double Cross: take all but 2 boxes in a long chain. it forces the opponent to open up another long chain for you, even if they get the 2 boxes Long Chain: make initial number of dots and numbers of long chains even if first player and odd if second player |
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Mancala |
-A game that originated in Africa during the 4th century -it is a partisan game, it is partially solved, and the first player has the winning strategy -many contributors. two of them were champion and cartenson. |
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Endgame |
-the final stage of a game when few pieces or cards remain -seen in go, pente and gomoku |
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John Nash |
-From USA, 20th century -invented Hex and proved that there is always a winner using the Brouwers fixed point theorem and that the first player has the winning strategy |
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hannah |
schwartz |
