The Pythagorean Theorem - Educational Outreach

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Epilogue: The Crown and the Jewels“Geometry has two great treasures;One is the Theorem of Pythagoras:The other, the division of a line into extreme and mean ratio.The first we may compare to a measure of gold;The second we may name a precious jewel.” Johannes KeplerSaint Paul says his First Epistle to the Corinthians,“For now we see through a glass, darkly…” Though thequote may be out of context, the thought freely standing onits own is carries the truth of the human condition.Humans are finite creatures limited by space, time, and theability of a three-pond mass to perceive the wondrousworkings of the Near Infinite…or Infinite. Even of that—Near Infinite or Infinite—we are not sure.Figure E.1: Beauty in Order166
Mathematics is one of the primary tools we humans use tosee through the glass darkly. With it, we can both discoverand describe ongoing order in the natural laws governingthe physical universe. Moreover, within that order we findcomfort, for order implies design and purpose—as opposedto chaos—through intelligence much grander than anythingproduced on a three-pound scale. The study of mathematicsas a study of order is one way to perceive cosmic-leveldesign and purpose. But no matter how good or howcomplete the human invention of mathematics can be, itstill allows just a glimpse through the darkened glass.Figure E.1, a mandala of sorts, hints at this truth oforder as it applies to the Pythagorean Theorem—which isbut a single pattern within the total synchronicity foundthroughout the cosmos. The four gradually shaded trianglesspeak of Pythagoras and one of the first Pythagorean proofs.Moreover, each triangle is golden via the ratio of hypotenuseto side, in deference to Kepler’s statement regarding the twogreat geometric treasures. Four also numbers the primarywaveforms in the visible spectrum—red, blue, green, andyellow— which can be mathematically modeled by advancedtrigonometric patterns (not addressed in this generalreadershipvolume). Since right triangles are intricatelylinked to circles, these two geometric figures are coupled inone-to-one fashion. A great circle, expressing unity,encompasses the pattern. White, as the synthesis of thevisual rainbow, appropriately colors all five circles. But evenwithin the beauty of pattern, linkage, and discovery asexpressed by our mandala, there is black. Let this darknessrepresent that which remains to be discovered or that whichis consigned to mystery. Either way, it becomes part of whowe are as humans as our innate finiteness seeks tocomprehend that which is much greater.It is my hope that this book has allowed a briefglimpse through the glass, a glimpse at a superbly simpleyet subtle geometric pattern called the PythagoreanTheorem that has endured for 4000 years or more ofhuman history.167
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The PythagoreanTheoremCrown Jewel o
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The Pythagorean TheoremCrown Jewel
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Table of ContentsList of Tables and
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List of Tables and FiguresTablesNum
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Figures…continuedNumber and Title
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List of Proofs and DevelopmentsSect
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PrefaceThe Pythagorean Theorem has
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1 : 2x2x2****22 13x 7 13x 7 0 : a
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1) Consider the Squares“If it was
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s090045 0ss45450450090Figure 1.3: O
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Notice that the right-triangle prop
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The middle square (minus the donut
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Our proof in Chapter 1 has been by
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2) Four Thousand Years of Discovery
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The proof Pythagoras is thought to
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cabcFigure 2.4: Annotated Square wi
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Figure 2.6 is the diagram for a sec
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1x:b2:Area(CBA)3:Area( ABC)Area(ABC
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Euclid’s proof of the Pythagorean
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First we establish that the two tri
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1 2y b b: y b c c2:A3:A4:Aunshaded
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XB'A090 CBFigure 2.12: Euclid’s C
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2.3) Liu Hui Packs the SquaresLiu H
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One of my favorites was Boxel, a ga
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2.4) Kurrah Transforms the Bride’
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Note: as is the occasional custom i
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2.5) Bhaskara Unleashes the Power o
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2.6) Leonardo da Vinci’s Magnific
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In Figure 2.24, we enlarge Step 5 a
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But, this is precisely the nature o
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2.8) Henry Perigal’s TombstoneHen
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In the century following Perigal, b
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Returning to Henry Perigal, Figure
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Figure 2.32 is President Garfield
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Even though the Cartesian coordinat
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B 2← γ →α y βx C-xC 2Figure
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Being polynomial in form, the funct
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Thus, the critical point ( C 2 ,0)i
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Sincexcpwas chosen on an arbitrary
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∂F/∂x = ∂F/∂y = 0x(C - x) 2
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Define a new function21 A2 A3 2xcp
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2.11) Shear, Shape, and AreaOur las
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Step 1: Cut the big square into two
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In Table 2.3, we briefly summarize
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We can formally state this similari
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What we will do in Section 3.2 is p
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There are three Pythagorean-triple
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Thus, if we multiply any given prim
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The last equality shows that the in
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A Pythagorean Quartet is a set of f
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3.5) Pythagoras and the Three Means
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yxy MHxab2ab2 y x FC y ab2ab 2ab y
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Hero was the founder of the Higher
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5 : Solve for area using the formul
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4 : Construct the parallelogram wit
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We already have introduced Thabit i
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Some historians believe Stewart’s
Page 117 and 118: In pre-computer days, both all four
Page 119 and 120: One example of a periodic process i
Page 121 and 122: The six trigonometric functions are
Page 123 and 124: The following four relationships ar
Page 125 and 126: As with any set of identities, the
Page 127 and 128: Figure 3.17 can be used as a jumpin
Page 129 and 130: The remaining two pillars are the L
Page 131 and 132: a2Law of Cosines c2 b2 2bccos()The
Page 133 and 134: An example of a Diophantine equatio
Page 136 and 137: However, Euler’s Conjecture did n
Page 138 and 139: 4) Pearls of Fun and WonderPearls o
Page 140 and 141: We have achieved five-digit accurac
Page 142 and 143: 140212213116)74370116)(74370(370)74
Page 144 and 145: X X X X O X X O X XO X X O O OO X O
Page 146 and 147: 4.3) Earth, Moon, Sun, and StarsIn
Page 148 and 149: The four-step solution follows. All
Page 150 and 151: Figure 4.9 depicts Eratosthenes’
Page 152 and 153: ---High above the earth, we see the
Page 154 and 155: The commonly accepted value is abou
Page 156 and 157: 1BC : tan( BAC)BABC BAtan(BAC)0BC
Page 158 and 159: BD ACtan( 21 )Even for the closest
Page 160 and 161: 3w 2h s w 5 s w 0.38196s&5 1s
Page 162 and 163: Figure 4.16: Triangular Phi---The h
Page 164 and 165: What remains to be done is to devel
Page 166 and 167: 3 : h32 4 h222 2 24C 2 h33162 2 2
Page 170 and 171: Not only has it endured, but the Py
Page 172 and 173: A] Greek AlphabetGREEK LETTERUpper
Page 174 and 175: C] Geometric FoundationsThe Paralle
Page 176 and 177: 8. Congruent Triangles: Two triangl
Page 178 and 179: 3. Rectangle: A bh : P 2b 2h, b &
Page 180 and 181: Recreational Mathematics11. Pickove
Page 182 and 183: GGolden RatioDefined and algebraica
Page 184 and 185: Puzzles (cont)Pythagorean Magic Squ
Page 186 and 187: Pythagorean Theorem ProofsAlgebraic
Page 188: Trigonometry (cont)Eratosthenes mea
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The Pythagorean Theorem - Educational Outreach

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