The Pythagorean Theorem - Educational Outreach

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X X X X O X X O X XO X X O O OO X O X O OO O O O O X O X X XX X X X X O X OX X O O X O O XO O X X O X O XO O O O O X X OX X O O O X X XX X X X X O O OO O X X X O X XO O O O O X O OFigure 4.3: 4X4 Magic PatternsThe 4X4 magic square has within it several sum-ofsquaresand sum-of-cubes equalities that provide additionalexamples of power sums as discussed in Section 3.8.1. The sum of the squares in the first row equals the sumof the squares in the fourth row: A similar equalityholds for the second and third rows.2 2 2 2 2 2 21 15 6 12 14 4 9 2. The sum of the squares in the first column equals thesum of the squares in the fourth column. A similarequality holds for the second and third columns.2 2 2 2 2 2 21 8 11141213 2 3. The sum of the cubes for the numbers on the twodiagonals equals the sum of the cubes for the numbersremaining within the square:3 31 10153 63163133 73 2314 914233 5 433 33123 311 8The reader is invited to verify all three power sums!77322
Magic squares of all types have intrigued mathenthusiasts for decades. What you see below in Figure 4.4is a Pythagorean masterpiece that couples three magicsquares of different sizes (or orders) with the truth of thePythagorean Theorem. Royal Vale Heath, a well-knownBritish puzzle maker, created this wonder in England priorto 1930. Of the fifty numbers used in total, none appearsmore than once.THE PYTHAGOREAN 3-4-5 WONDER SET OFTHREE MAGIC SQUARES3X3 magicsum is 174.Square thesum of ninenumbers toobtain272,4845X5 magic sum is also4X4 magic sum is174. Square the sum ofalso 174. Squaretwenty-five numbers tothe sum of sixteenobtain 756,900numbers toobtain 484,416 16 22 28 34 7436 43 48 47 33 73 20 21 2761 54 59 49 46 37 42 25 26 32 72 1956 58 60 39 40 51 44 71 18 24 30 3157 62 55 50 45 38 41 29 35 70 17 233 2 2 2 4 5 & 272,484484,416 756, 900 !Figure 4.4: Pythagorean Magic Squares143
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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
Page 93 and 94: What we will do in Section 3.2 is p
Page 95 and 96: There are three Pythagorean-triple
Page 97 and 98: Thus, if we multiply any given prim
Page 99 and 100: The last equality shows that the in
Page 101 and 102: A Pythagorean Quartet is a set of f
Page 103 and 104: 3.5) Pythagoras and the Three Means
Page 105 and 106: yxy MHxab2ab2 y x FC y ab2ab 2ab y
Page 107 and 108: Hero was the founder of the Higher
Page 109 and 110: 5 : Solve for area using the formul
Page 111 and 112: 4 : Construct the parallelogram wit
Page 113 and 114: We already have introduced Thabit i
Page 115 and 116: 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 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 168 and 169: Epilogue: The Crown and the Jewels
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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