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S PHERE P ACKING Math Day 2015 Kristin DeVleming

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M OTIVATION If I have a big box, how many oranges can I fit in it? How do I arrange the oranges to get the most in the box?

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W HAT IS S PHERE P ACKING ? Arrangement of non- overlapping spheres in some containing space Types: Equal Unequal Regular Irregular

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S PHERE P ACKING How would you get the most oranges in the box? “Densest” sphere packing?

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S PHERE P ACKING

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“Face Centered Cubic” (FCC) What is the density of FCC?

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S PHERE P ACKING 6 half spheres (one on each face) 8 1/8 th spheres (one on each corner) Total = 4 spheres

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S PHERE P ACKING

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Is this the best we can do???

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S PHERE P ACKING

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hexagonal close packing face centered cubic HCP and FCC have the same density!

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S PHERE P ACKING Kepler Conjecture: No packing of spheres of the same radius has density greater than the face-centered cubic packing.

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H ISTORY Kepler (1611): The Six-Cornered Snowflake Conjectured FCC was densest packing Gauss (1831): Proved this was densest lattice packing Hales (1998): Proved this was densest out of all packings 2006: checked proof with automated proof checking

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M ORE Q UESTIONS Can we prove this without using a computer? Can we make sense of sphere packing in other dimensions? What about unequal sphere packing? WHY DO WE CARE?

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A PPLICATIONS Matter is made up of atoms which are roughly spherical Crystals are made up of atoms arranged in a repeated pattern

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Diamond A PPLICATIONS Graphite

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A PPLICATIONS Graphite and diamond have the same chemical structure (C), but different sphere packing arrangements

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A PPLICATIONS Graphite has its atoms arranged is hexagonal sheets Sheets can move from side to side: Easy to break “Sea of electrons” between layers: Conducts electricity

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A PPLICATIONS Diamond has its atoms arranged in a tetrahedral pattern Each atom has 4 neighbors: No free electrons, insulator To move one atom, must move the surrounding ones: Very hard

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A PPLICATIONS Crystallography: determining how atoms are arranged in a crystal

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A PPLICATIONS We can identify sphere packing structures with crystallography techniques

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A PPLICATIONS Error Correcting Codes

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A PPLICATIONS Assign each letter a “code word” Make sure code words have at least 2r differences code word: 110 point (1,1,0); center of sphere with radius r

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A PPLICATIONS code word: 110 point (1,1,0); center of sphere with radius r Each code word is in a (unique) sphere, spheres don’t overlap If we make less than r errors, the code word with errors is still in the same sphere, so … If the code word is sent with less than r errors, we can correct it!

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S PHERE P ACKING Simple questions, hard answers Real world applications

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M ORE Q UESTIONS Can we do “sphere packing” with other shapes? Where else does sphere packing appear in the “real world”? Can we say anything about random sphere packing?

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M ORE Q UESTIONS What questions do YOU have?

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Gold exhibits a face-centered cubic unit cell that is 4.08 A on a side. Estimate gold’s density, in g/cm 3. D = m V = 4 atoms (197.0 amu/at.) (4.08 A)

Gold exhibits a face-centered cubic unit cell that is 4.08 A on a side. Estimate gold’s density, in g/cm 3. D = m V = 4 atoms (197.0 amu/at.) (4.08 A)

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