The Pythagorean Theorem: Crown Jewel of Mathematics
John C. Sparks
Format: PDF / Kindle (mobi) / ePub
The Pythagorean Theorem, Crown Jewel of Mathematics chronologically traces the Pythagorean Theorem from a conjectured beginning, Consider the Squares (Chapter 1), through 4000 years of Pythagorean proofs, Four Thousand Years of Discovery (Chapter 2), from all major proof categories, 20 proofs in total. Chapter 3, Diamonds of the Same Mind, presents several mathematical results closely allied to the Pythagorean Theorem
along with some major Pythagorean “spin-offs” such as Trigonometry. Chapter 4, Pearls of Fun and Wonder, is a potpourri of classic puzzles, amusements, and applications.
An Epilogue, The Crown and the Jewels, summarizes the importance of the Pythagorean Theorem and suggests paths for further exploration. Four appendices service the reader: A] Greek Alphabet, B] Mathematical Symbols, C] Geometric Foundations, and D] References. For the reader who may need a review of elementary geometric concepts before engaging this book, Appendix C is highly recommended. A Topical Index completes the book.
Chair 2.17: Packing the Bride’s Chair into the Big Chair 2.18: Kurrah’s Operation Transformation 2.19: The Devil’s Teeth 2.20: Truth versus Legend 2.21: Bhaskara’s Real Power 2.22: Leonardo da Vinci’s Symmetry Diagram 2.23: Da Vinci’s Proof in Sequence 2.24: Subtle Rotational Symmetry 2.25: Legendre’s Diagram 2.26: Barry Sutton’s Diagram 2.27: Diagram on Henry Perigal’s Tombstone 2.28: Annotated Perigal Diagram 2.29: An Example of Pythagorean Tiling 2.30: Four Arbitrary Placements… 2.31: Exposing
Geometric Foundations, and D] References. For the reader who may need a review of elementary geometric concepts before engaging this book, Appendix C is highly recommended. A Topical Index completes the book. A Word on Formats and Use of Symbols One of my interests is poetry, having written and studied poetry for several years now. If you pick up a textbook on poetry and thumb the pages, you will see poems interspersed between explanations, explanations that English professors will call prose.
exhibited by the functions. five remaining trigonometric In general, trigonometric functions cycle through the same values over and over again as the independent variable t indefinitely increases on the interval [0, ) or, by the same token, indefinitely decreases on the interval (,0] , corresponding to repeatedly revolving around the rim of the unit circle in Figure 3.15. 119 This one characteristic alone suggests that (at least in theory) trigonometric functions can be used to model the
student is left to believe that this is how the sequence actually happened in a historical context. Recall that Thomas Edison had four-thousand failures before finally succeeding with the light bulb. Mathematicians are no less prone to dead ends and frustrations! Since the sum of any two acute angles in any one of 0 the right triangles is again 90 , the lighter-shaded figure bounded by the four darker triangles (resulting from Step 4) is a square with area double that of the original square.
ab) The second step is to equate these expressions and algebraically simplify. 2 set : (a b) 2 c 2 4( 12 ab) a 2 2ab b 2 c 2 2ab a2 b2 c2 31 Notice how quickly and easily our result is obtained once algebraic is used to augment the geometric picture. Simply put, algebra coupled with geometry is superior to geometry alone in quantifying and tracking the diverse and subtle relationships between geometric whole and the assorted pieces. Hence, throughout the remainder of