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Problem : Can 5 test tubes be spun simultaneously in a 12-hole centrifuge?

Problem : Can 5 test tubes be spun simultaneously in a 12-hole centrifuge?. No vector calculus / trig! No equations! Truth is guaranteed! Fundamental principles exposed! Easy to generalize! High elegance / beauty!. What does “ balanced ” means? Why are 3 test tubes balanced? Symmetry !

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Problem : Can 5 test tubes be spun simultaneously in a 12-hole centrifuge?

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  1. Problem: Can 5 test tubes be spun simultaneously in a 12-hole centrifuge? No vector calculus / trig! No equations! Truth is guaranteed! Fundamental principles exposed! Easy to generalize! High elegance / beauty! • What does “balanced” means? • Why are 3 test tubes balanced? • Symmetry! • Can you merge solutions? • Superposition! • Linearity! ƒ(x + y) = ƒ(x) + ƒ(y) • Can you spin 7 test tubes? • Complementarity! • Empirical testing…

  2. 1 1 Problem: Given any five points in/on the unitsquare, is there always a pair with distance ≤ ? • What approaches fail? • What techniques work and why? • Lessons and generalizations

  3. 1 1 1 Problem: Given any five points in/on the unit equilateral triangle, is there always a pair with distance ≤ ½ ? • What approaches fail? • What techniques work and why? • Lessons and generalizations

  4. … X X X X X = 2 Problem: Solve the following equation for X: where the stack of exponentiated x’s extends forever. • What approaches fail? • What techniques work and why? • Lessons and generalizations

  5. Problem: For the given infinite ladder of resistors of resistance R each, what is the resistance measured between points x and y? x y • What approaches fail? • What techniques work and why? • Lessons and generalizations

  6. Historical Perspectives • Georg Cantor (1845-1918) • Created modern set theory • Invented trans-finite arithmetic • (highly controvertial at the time) • Invented diagolanization argument • First to use 1-to-1 correpondences with sets • Proved some infinities “bigger” than others • Showed an infinite hierarchy of infinities • Formulated continuum hypothesis • Cantor’s theorem, “Cantor set”, Cantor dust, • Cantor cube, Cantor space, Cantor’s paradox • Laid foundation for computer science theory • Influenced Hilbert, Godel, Church, Turing

  7. Problem: How can a new guest be accommodated in a full infinite hotel? ƒ(n) = n+1

  8. Problem: How can an infinity of new guests be accommodated in a full infinite hotel? ƒ(n) = 2n …

  9. Problem: How can an infinity of infinities of new guests be accommodated in a full infinite hotel? … 15 one-to-one correspondence 10 14 6 9 13 5 8 3 12 4 11 7 2 1

  10. Problem: Are there more integers than natural #’s? ℕ Ìℤ ℕ ¹ℤ So |ℕ|<|ℤ|? Rearrangement: Establishes 1-1 correspondence ƒ: ℕ « ℤ Þ|ℕ|=|ℤ| ℤ -4 -4 -3 -3 -2 -2 -1 -1 0 0 1 1 2 2 3 3 4 4 ℤ 1 2 3 4 5 6 7 8 9 ℕ

  11. Problem: Are there more rationals than natural #’s? … 7 … 6 … 5 … 4 … 3 … 3 7 1 2 1 3 7 4 6 6 7 4 1 5 3 5 1 2 2 2 7 3 6 1 4 2 2 5 7 1 3 1 2 6 7 4 4 4 6 5 6 6 7 7 2 1 3 5 3 5 4 4 6 5 3 5 2 5 2 2 1 1 1 1 1 1 3 3 1 5 5 5 5 5 5 4 3 4 4 4 4 4 4 2 2 8 2 8 7 6 8 8 6 7 8 6 7 7 7 8 8 2 6 3 3 3 6 3 7 2 7 6 6 … 1 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 ℕ Ìℚ ℕ ¹ℚ So |ℕ|<|ℚ|? Dovetailing: Establishes 1-1 correspondence ƒ: ℕ «ℚ Þ|ℕ|=|ℚ| 55 17 18 19 20 21 16 15 14 13 22 5 6 7 12 23 28 8 4 3 27 11 24 26 9 1 2 10 25 …

  12. Problem: Are there more rationals than natural #’s? 1 4 1 5 7 6 4 5 3 2 1 3 5 2 4 6 7 4 7 5 3 1 2 4 6 5 6 1 1 3 7 2 1 2 3 5 7 4 6 6 7 4 3 6 2 2 2 7 6 1 7 5 3 3 5 4 1 5 1 2 1 1 1 2 3 3 1 2 4 5 5 4 4 4 4 5 4 3 3 4 2 3 2 1 2 7 8 7 8 8 8 6 7 7 8 5 6 8 5 7 6 7 6 8 2 6 7 6 6 5 3 3 24 25 26 27 28 29 39 … ℕ Ìℚ ℕ ¹ℚ So |ℕ|<|ℚ|? Dovetailing: Establishes 1-1 correspondence ƒ: ℕ «ℚ Þ|ℕ|=|ℚ| 7 30 23 22 … 6 12 13 14 15 38 31 21 … 5 Avoiding duplicates! 11 10 32 16 … 4 4 5 33 37 9 17 … 3 6 3 18 34 … 2 35 36 20 7 1 2 8 19 … 1 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 …

  13. Problem: Are there more rationals than natural #’s? 4 5 3 1 2 7 6 4 3 1 2 5 1 1 4 5 2 3 6 3 4 7 7 6 2 6 7 2 1 2 4 5 3 1 2 7 6 4 5 3 1 2 6 6 7 5 4 6 7 5 3 3 7 5 4 1 4 4 2 3 4 2 3 3 3 2 4 4 5 2 5 5 2 4 5 2 5 5 5 1 3 4 3 2 7 6 6 6 1 6 6 1 1 7 1 8 8 8 1 8 8 7 8 1 8 7 7 3 7 6 7 6 21 26 … ℕ Ìℚ ℕ ¹ℚ So |ℕ|<|ℚ|? Dovetailing: Establishes 1-1 correspondence ƒ: ℕ «ℚ Þ|ℕ|=|ℚ| 7 17 … 6 11 16 20 25 … 5 9 15 24 … 4 5 8 14 19 … 3 7 3 13 23 … 2 18 22 12 4 1 2 6 10 … 1 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 …

  14. Problem: Why doesn’t this “dovetailing” work? 4 5 3 1 2 7 6 4 3 1 1 7 5 1 1 4 5 2 3 6 3 4 2 7 6 6 7 2 1 2 6 4 5 3 2 7 6 4 5 3 1 2 7 3 7 5 4 6 5 7 3 2 5 4 6 1 4 4 2 4 2 3 4 2 4 3 4 4 3 2 3 5 5 3 5 5 5 2 5 5 3 6 2 2 7 1 3 6 1 6 6 6 1 7 6 6 7 1 7 7 1 1 7 8 8 8 8 8 8 8 7 1 … There’s no “last” element on the first line! So the 2nd line is never reached! Þ 1-1 function is not defined! 7 … 6 … 5 … 4 … 3 … 2 … 1 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 …

  15. Dovetailing Reloaded Dovetailing: ƒ:ℕ «ℤ -4 -4 -3 -3 -2 -2 -1 -1 0 0 1 1 2 2 3 3 4 4 ℤ 1 2 3 4 5 6 7 8 9 ℕ • To show |ℕ|=|ℚ| we can construct ƒ:ℕ«ℚ by sorting x/y • by increasing key max(|x|,|y|), while avoiding duplicates: • max(|x|,|y|) = 0 : {} • max(|x|,|y|) = 1: 0/1, 1/1 • max(|x|,|y|) = 2: 1/2, 2/1 • max(|x|,|y|) = 3: 1/3, 2/3, 3/1, 3/2 • . . . {finite new set at each step} • Dovetailing can have many disguises! • So can diagonalization! Dovetailing! 1 2 3 4 5 6 7 8

  16. Theorem: There are more reals than rationals / integers. Proof [Cantor]: Assume a 1-1 correspondence ƒ: ℕ « ℝ i.e., there exists a table containing all of ℕ andall ofℝ: ℕ ℝ Diagonalization Non-existence proof! . . . Îℝ X = 0 . 2 1 9 3 4 • But X is missing from our table! X¹ ƒ(k) " kÎℕ • Þ ƒ not a 1-1 correspondence • Þcontradiction • ℝ is not countable! There are more reals than rationals / integers!

  17. Problem 1: Why not just insert X into the table? Problem 2: What if X=0.999… but 1.000… is already in table? ℕ ℝ Diagonalization Non-existence proof! . . . Îℝ X = 0 . 2 1 9 3 4 • Table with X inserted will have X’ still missing! • Inserting X (or any number of X’s) will not help! • To enforce unique table values, we can avoid • using 9’s and 0’s in X.

  18. Non-Existence Proofs • Must cover all possible (usually infinite) scenarios! • Examples / counter-examples are not convincing! • Not “symmetric” to existence proofs! Ex: proof that you are a millionare: “Proof” that you arenot a millionare ? Non-existence proofs are often hard! Existence proofs can be easy! P¹NP

  19. Cantor set: • Start with unit segment • Remove (open) middle third • Repeatrecursively on all remaining segments • Cantor set is all the remaining points Total length removed: 1/3 + 2/9 + 4/27 + 8/81 + … = 1 Cantor set does not contain any intervals Cantor set is not empty (since, e.g. interval endpoints remain) An uncountable number of non-endpoints remain as well (e.g., 1/4) Cantor set is totally disconnected (no nontrivial connected subsets) Cantor set is self-similar with Hausdorff dimension of log32=1.585 Cantor set is a closed, totally bounded, compact, complete metric space, with uncountable cardinality and lebesque measure zero

  20. Cantor dust (2D generalization): Cantor set crossed with itself

  21. Cantor cube (3D): Cantorset crossed with itselfthree times

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