1 / 26

Elegant Solutions: Exploring Centrifuge, Geometry, and Infinite Ladders

Dive into fundamental principles without complex equations. Discover symmetry, superposition, and infinite hierarchies elegantly explored. Explore geometric problems and infinite ladder resistance calculations.

smithmartin
Download Presentation

Elegant Solutions: Exploring Centrifuge, Geometry, and Infinite Ladders

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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” mean? • 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 diagonalization argument • First to use 1-to-1 correspondences 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 millionaire: “Proof” that you arenot a millionaire ? 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

More Related