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On Formulations of Pythagorean Theorem by Parker Emmerson Why this can help us prove string theory? Within even the smallest unit of time, there is an implied ten dimensionality. Because it is mathematically proven, it is no longer theoretical. The mathematics is directly correlated to the perceptual experience just as the geometer measures as he perceives.

Preface : r ^ 2 = h ^ 2 + r1 ^ 2 q r = 2 p r - 2 p r1 = 2 p r - 2 p qr -2pr = - 2p qr -2pr 2p qr -2pr 2p qr -2pr 2p

=

Hr ^ 2 - h ^ 2L

Hr ^ 2 - h ^ 2L

Hr ^ 2 - h ^ 2L ^ 2 = Hr ^ 2 - h ^ 2L

^ 2 + h ^ 2 = r ^ 2 = h ^ 2 + r1 ^ 2

Relevant Lemmas to the Formulation The height of the cone can be calculated in terms of only r and q, thus b is a function of q alone.

Lemma 4

Proof. Since we have shown that q r = 2 p r - 2 p r 1 and r1 Ă˜ r2 - h2 , we can substitute the expression for r1 , calculated from the Pythagorean theorem in terms of the height of the cone and the initial radius of the circle, into the expression for q r in terms of the change in circumference of the initial circle to the circle, which is the base of the cone. q r = 2 p r - 2 p 4 p r2 q-r2 q2 2p

Hr ^ 2 - h ^ 2L , thus, h =

q=2 pÂą

p2 - p2 Sin@bD2

2ph

= (r Sin[b]). From

= r, we note that r =

4 p q-q

2 p r2 +

=

r4 -r2 h2 r2

, because 1 =

2

2 p r Sin@bD

, so

4 p q-q2

2 p Sin@bD 4 p q-q2

Lemma 6 The height of the cone can be calculated in terms of only r and q, thus q is a function of b alone, and the initial radius can be calculated purely in terms of the angle q. b = ArcSinB

H4 p - qL q

Sin@bD =

2p h r

=

F

4 p r2 q - r2 q2 r2p

=

4 p r2 q - r2 q2 4 p2 r

=

r H4 p - qL q 4 p2

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2

Pythagorean Theorem Roots by Parker Emmerson © 2009-2010.nb

b Ø ArcSinB

4 p r q - r q2 4 p2

SolveBArcSinB

4 p r q - r q2 4p

::r Ø

2p

H4 p - qL q

F = ArcSinB

2

F

2p

F == ArcSinB

H4 p - qL q 2p

F, rF

H4 p - qL q H4 p - qL q

>>

ü A quick calculation: 2 p r2 +

2p

4p-

r4 -r2 h2

2 p r2 +

r2

r4 -r2 h2 r2

SolveBr ==

, hF 2 p r2 +

4p-

r4 -r2 h2

2 p r2 +

r2

r4 -r2 h2 r2

88h Ø - 1<, 8h Ø 1<<

2 p r2 +

2p

4p-

r4 -r2 h2

2 p r2 +

r2

r4 -r2 h2 r2

SolveBr ==

, rF 2 p r2 +

4p-

r4 -r2 h2

2 p r2 +

r2

r4 -r2 h2 r2

8<

2p

p2 - p2 Sin@bD2

4p-2 p+

p2 - p2 Sin@bD2

2 p+

SolveBr ==

, bF 4p-2 p+

2

2

p - p Sin@bD

2

2 p+

2

2

p - p Sin@bD

1 1 ::b Ø - ArcSinB F>, :b Ø ArcSinB F>> r r

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2


Pythagorean Theorem Roots by Parker Emmerson © 2009-2010.nb

2p

4p-2 p+

p2 - p2 Sin@bD2

2 p+

p2 - p2 Sin@bD2

SolveBr ==

, hF 2 p r2 +

4p1

::h Ø -

r4 -r2 h2

2 p+

r2

Csc@bD - K- 2 p + p Sin@bD2 + 2 p

p2 - p2 Sin@bD

2

- H- 1 + Sin@bDL H1 + Sin@bDL +

2 r

p - KKp + p 2 r

1

- H- 1 + Sin@bDL H1 + Sin@bDL OOO -

- H- 1 + Sin@bDL H1 + Sin@bDL

- KKp + p :h Ø

- H- 1 + Sin@bDL H1 + Sin@bDL O K4 p - 2 Kp + p

- H- 1 + Sin@bDL H1 + Sin@bDL O K4 p - 2 Kp + p

Csc@bD - K- 2 p + p Sin@bD2 + 2 p

- H- 1 + Sin@bDL H1 + Sin@bDL OOOO>,

- H- 1 + Sin@bDL H1 + Sin@bDL +

2 r

p - KKp + p 2 r

- H- 1 + Sin@bDL H1 + Sin@bDL O K4 p - 2 Kp + p

- H- 1 + Sin@bDL H1 + Sin@bDL OOO -

- H- 1 + Sin@bDL H1 + Sin@bDL

- KKp + p

RevolutionPlot3DB-

- H- 1 + Sin@bDL H1 + Sin@bDL O K4 p - 2 Kp + p 1

Csc@bD - K- 2 p + p Sin@bD2 + 2 p

- H- 1 + Sin@bDL H1 + Sin@bDL OOOO>>

- H- 1 + Sin@bDL H1 + Sin@bDL +

p 2 r - KKp + p 2 r

- H- 1 + Sin@bDL H1 + Sin@bDL O K4 p - 2 Kp + p

- H- 1 + Sin@bDL H1 + Sin@bDL - KKp + p

K4 p - 2 Kp + p

- H- 1 + Sin@bDL H1 + Sin@bDL OOO -

- H- 1 + Sin@bDL H1 + Sin@bDL O

- H- 1 + Sin@bDL H1 + Sin@bDL OOOO, 8r, - 1, 1<, 8b, - p, p<F

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3


4

Pythagorean Theorem Roots by Parker Emmerson Š 2009-2010.nb

RevolutionPlot3DB:

1

Csc@bD - K- 2 p + p Sin@bD2 + 2 p

- H- 1 + Sin@bDL H1 + Sin@bDL +

p 2 r - KKp + p 2 r

1

- H- 1 + Sin@bDL H1 + Sin@bDL OOO -

- H- 1 + Sin@bDL H1 + Sin@bDL

- KKp + p -

- H- 1 + Sin@bDL H1 + Sin@bDL O K4 p - 2 Kp + p

- H- 1 + Sin@bDL H1 + Sin@bDL O K4 p - 2 Kp + p

Csc@bD - K- 2 p + p Sin@bD2 + 2 p

- H- 1 + Sin@bDL H1 + Sin@bDL OOOO,

- H- 1 + Sin@bDL H1 + Sin@bDL +

p 2 r - KKp + p 2 r

- H- 1 + Sin@bDL H1 + Sin@bDL O K4 p - 2 Kp + p

- H- 1 + Sin@bDL H1 + Sin@bDL - KKp + p

K4 p - 2 Kp + p

- H- 1 + Sin@bDL H1 + Sin@bDL OOO -

- H- 1 + Sin@bDL H1 + Sin@bDL O

- H- 1 + Sin@bDL H1 + Sin@bDL OOOO>, 8r, - 1, 1<, 8b, - p, p<F

The Formulation of New Pythagorean Expressions from Difference in Circumferences Equals an Arc Length of the Initial Circle (Which is Folded into a Cone) Applied to Pythagorean Theorem Printed by Mathematica for Students


Pythagorean Theorem Roots by Parker Emmerson © 2009-2010.nb

The Formulation of New Pythagorean Expressions from Difference in Circumferences Equals an Arc Length of the Initial Circle (Which is Folded into a Cone) Applied to Pythagorean Theorem r ^ 2 = h ^ 2 + r1 ^ 2

2p

H4 p - qL q H4 p - qL q

2p

H4 p - qL q H4 p - qL q

^ 2 = h ^ 2 + r1 ^ 2

4 p r2 q - r2 q2

r^2 =

4 p r2 q - r2 q2

^ 2 + r1 ^ 2 =

2p

^2 +

2p

qr -2pr 2p

^2

ü Substitutions that Cancel

SolveBr ^ 2 ==

4 p r2 q - r2 q2 2p

^2 +

qr -2pr 2p

^ 2, rF

88<<

SolveBr ^ 2 ==

4 p r2 q - r2 q2 2p

^2 +

qr -2pr 2p

^ 2, qF

88<<

SolveB

2p

4p

H4 p - qL q

^ 2 ==

H4 p - qL q

2p

H4 p-qL q H4 p-qL q

2

q-

2p

H4 p-qL q

2

q2

H4 p-qL q

q

2p

^2 +

2p

H4 p-qL q

-2p

H4 p-qL q

2p

H4 p-qL q H4 p-qL q

2p

88<<

ü Substitutions that Don't Cancel

SolveBr ^ 2 ==

4 p HrL2 q - HrL2 q2 2p 2 p r2 -

:8q Ø 2 p<, 8q Ø 2 p<, :q Ø

SolveBr ^ 2 ==

4 p HrL2 q - HrL2 q2 2p

q ^2 +

2p

H4 p-qL q

-2p

H4 p-qL q

2 p r2 + >, :q Ø

qr -2p ^2 +

H4 p-qL q H4 p-qL q

2p

- r2 + r4 r2

2p

2p

- r2 + r4 r2

H4 p-qL q H4 p-qL q

2p

Printed by Mathematica for Students

^ 2, qF

^ 2, qF

>>

^ 2, qF

5


6

Pythagorean Theorem Roots by Parker Emmerson © 2009-2010.nb

2 p r2 ::q Ø

:q Ø

- r2 + r4 >, :q Ø

r2 3p 2

-

1 2

2 p r2 +

- r2 + r4 >,

r2

p2 I16 r2 + 3 r4 M

3 p2 +

1ê3

3

1ê3

r

2

4

6

216 r - 9 r + 8

3

6

8

- 64 r + 207 r - 27 r

+

10

1ê3

p2 216 r4 - 9 r6 + 8

- 64 r6 + 207 r8 - 27 r10

3

-

32ê3 r2

1

p2 I16 r2 + 3 r4 M

6 p2 -

2

1ê3

31ê3 r2 216 r4 - 9 r6 + 8

-

- 64 r6 + 207 r8 - 27 r10

3

1ê3

p2 216 r4 - 9 r6 + 8

- 64 r6 + 207 r8 - 27 r10

3

-

32ê3 r2

I8 p3 M ì

p2 I16 r2 + 3 r4 M

3 p2 +

1ê3

3

1ê3

r

2

4

6

216 r - 9 r + 8

3

6

8

- 64 r + 207 r - 27 r

10

1ê3

p2 216 r4 - 9 r6 + 8

3

- 64 r6 + 207 r8 - 27 r10 >,

32ê3 r2

:q Ø

3p 2

-

1 2

p2 I16 r2 + 3 r4 M

3 p2 +

1ê3

31ê3 r2 216 r4 - 9 r6 + 8

3

- 64 r6 + 207 r8 - 27 r10 1ê3

p2 216 r4 - 9 r6 + 8

3

- 64 r6 + 207 r8 - 27 r10 32ê3 r2

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+

+

+


Pythagorean Theorem Roots by Parker Emmerson © 2009-2010.nb

1

p2 I16 r2 + 3 r4 M

6 p2 -

2

1ê3

3

1ê3

r

2

4

6

216 r - 9 r + 8

6

8

- 64 r + 207 r - 27 r

3

-

10

1ê3

p2 216 r4 - 9 r6 + 8

- 64 r6 + 207 r8 - 27 r10

3

-

32ê3 r2

I8 p3 M ì

p2 I16 r2 + 3 r4 M

3 p2 +

1ê3

3

1ê3

r

2

4

6

216 r - 9 r + 8

3

6

8

- 64 r + 207 r - 27 r

+

10

1ê3

p2 216 r4 - 9 r6 + 8

- 64 r6 + 207 r8 - 27 r10

3

>,

32ê3 r2

:q Ø

3p 2

+

1 2

p2 I16 r2 + 3 r4 M

3 p2 +

1ê3

3

1ê3

r

2

4

6

216 r - 9 r + 8

6

8

- 64 r + 207 r - 27 r

3

+

10

1ê3

p2 216 r4 - 9 r6 + 8

- 64 r6 + 207 r8 - 27 r10

3

-

32ê3 r2

1 2

p2 I16 r2 + 3 r4 M

6 p2 -

1ê3

3

1ê3

r

2

4

6

216 r - 9 r + 8

3

6

8

- 64 r + 207 r - 27 r

-

10

1ê3

p2 216 r4 - 9 r6 + 8

- 64 r6 + 207 r8 - 27 r10

3

+

32ê3 r2

I8 p3 M ì

p2 I16 r2 + 3 r4 M

3 p2 +

1ê3

3

1ê3

r

2

4

6

216 r - 9 r + 8

3

6

8

- 64 r + 207 r - 27 r

>,

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10

+

7


8

Pythagorean Theorem Roots by Parker Emmerson © 2009-2010.nb

1ê3

p2 216 r4 - 9 r6 + 8

- 64 r6 + 207 r8 - 27 r10

3

>,

32ê3 r2

:q Ø

3p 2

+

1 2

p2 I16 r2 + 3 r4 M

3 p2 +

1ê3

3

1ê3

r

2

4

6

216 r - 9 r + 8

3

6

8

- 64 r + 207 r - 27 r

+

10

1ê3

p2 216 r4 - 9 r6 + 8

- 64 r6 + 207 r8 - 27 r10

3

+

32ê3 r2

1 2

p2 I16 r2 + 3 r4 M

6 p2 -

1ê3

31ê3 r2 216 r4 - 9 r6 + 8

3

-

- 64 r6 + 207 r8 - 27 r10 1ê3

p2 216 r4 - 9 r6 + 8

- 64 r6 + 207 r8 - 27 r10

3

+

32ê3 r2

I8 p3 M ì

p2 I16 r2 + 3 r4 M

3 p2 +

1ê3

3

1ê3

r

2

4

6

216 r - 9 r + 8

3

6

8

- 64 r + 207 r - 27 r 1ê3

p2 216 r4 - 9 r6 + 8

3

- 64 r6 + 207 r8 - 27 r10 32ê3 r2

Printed by Mathematica for Students

>>

10

+


Pythagorean Theorem Roots by Parker Emmerson © 2009-2010.nb

PlotB

3p 2

+

1

p2 I16 r2 + 3 r4 M

3 p2 +

2

1ê3

31ê3 r2 216 r4 - 9 r6 + 8

3

+

- 64 r6 + 207 r8 - 27 r10 1ê3

p2 216 r4 - 9 r6 + 8

- 64 r6 + 207 r8 - 27 r10

3

+

32ê3 r2

1 2

p2 I16 r2 + 3 r4 M

6 p2 -

1ê3

3

1ê3

r

2

4

6

216 r - 9 r + 8

3

6

8

- 64 r + 207 r - 27 r

-

10

1ê3

p2 216 r4 - 9 r6 + 8

- 64 r6 + 207 r8 - 27 r10

3

+

32ê3 r2

I8 p3 M ì

p2 I16 r2 + 3 r4 M

3 p2 +

1ê3

31ê3 r2 216 r4 - 9 r6 + 8

3

- 64 r6 + 207 r8 - 27 r10 1ê3

p2 216 r4 - 9 r6 + 8

3

- 64 r6 + 207 r8 - 27 r10 , 8r, - 1, 1<F

32ê3 r2

12.0

11.5

11.0

10.5

-1.0

-0.5

0.5

1.0

Note that r is to the 10 th dimension.

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+

9


10

Pythagorean Theorem Roots by Parker Emmerson © 2009-2010.nb

2p

4p

::q Ø

- 5 r2 + r4

- r2 + r4

2 p r2 +

4 p HrL2 q SolveBr ^ 2 ==

2p

q2

qr -2p ^2 +

2 p r2 + >, :q Ø

r2

>>

2

H4 p-qL q

- r2 + r4

>,

- r2 + r4

2p

2 p r2 -

^ 2, qF

- 5 r2 + r4

r2

H4 p-qL q

H4 p-qL q H4 p-qL q

4 + r2

>, :q Ø

r2

2p

2p

2 p - 2 + r2 + >, :q Ø

4 + r2 2 p r2 -

qr -2p ^2 +

2p

2 p - 2 + r2 -

::q Ø

q - HrL2 q2

H4 p-qL q

SolveBr ^ 2 ==

:q Ø

2

H4 p-qL q

2p

H4 p-qL q H4 p-qL q

2p

^ 2, qF

- r2 + r4 >,

r2

9q Ø RootA64 p6 - 64 p5 r2 Ò1 + I- 32 p4 + 144 p4 r2 M Ò12 64 p3 r2 Ò13 + I4 p2 - 24 p2 r2 M Ò14 + 4 p r2 Ò15 + r2 Ò16 &, 1E=, 9q Ø RootA64 p6 - 64 p5 r2 Ò1 + I- 32 p4 + 144 p4 r2 M Ò12 - 64 p3 r2 Ò13 + I4 p2 - 24 p2 r2 M Ò14 + 4 p r2 Ò15 + r2 Ò16 &, 2E=, 9q Ø RootA64 p6 - 64 p5 r2 Ò1 + I- 32 p4 + 144 p4 r2 M Ò12 - 64 p3 r2 Ò13 + I4 p2 - 24 p2 r2 M Ò14 + 4 p r2 Ò15 + r2 Ò16 &, 3E=, 9q Ø RootA64 p6 - 64 p5 r2 Ò1 + I- 32 p4 + 144 p4 r2 M Ò12 - 64 p3 r2 Ò13 + I4 p2 - 24 p2 r2 M Ò14 + 4 p r2 Ò15 + r2 Ò16 &, 4E=, 9q Ø RootA64 p6 - 64 p5 r2 Ò1 + I- 32 p4 + 144 p4 r2 M Ò12 - 64 p3 r2 Ò13 + I4 p2 - 24 p2 r2 M Ò14 + 4 p r2 Ò15 + r2 Ò16 &, 5E=, 9q Ø RootA64 p6 - 64 p5 r2 Ò1 + I- 32 p4 + 144 p4 r2 M Ò12 - 64 p3 r2 Ò13 + I4 p2 - 24 p2 r2 M Ò14 + 4 p r2 Ò15 + r2 Ò16 &, 6E=> PlotA RootA64 p6 - 64 p5 r2 Ò1 + I- 32 p4 + 144 p4 r2 M Ò12 - 64 p3 r2 Ò13 + I4 p2 - 24 p2 r2 M Ò14 + 4 p r2 Ò15 + r2 Ò16 &, 4E, 8r, - 100, 100<E 12.5660 12.5655 12.5650 12.5645 12.5640 12.5635

-100

-50

50

100

Printed by Mathematica for Students


Pythagorean Theorem Roots by Parker Emmerson © 2009-2010.nb

RevolutionPlot3DB 6

RootB64 p - 64 p

4 p2 - 24 p2

5

2p

2

H4 p - qL q

4

Ò1 + - 32 p + 144 p

H4 p - qL q 2p

H4 p - qL q H4 p - qL q

4

2p

H4 p - qL q H4 p - qL q

2

Ò12 - 64 p3 r2 Ò13 +

2

Ò14 + 4 p r2 Ò15 + r2 Ò16 &, 4F, 8r, - 100, 100<, 8q, - 2 p, 2 p<F

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11


12

Pythagorean Theorem Roots by Parker Emmerson © 2009-2010.nb

4 p HrL2 q SolveBr ^ 2 ==

H4 p-qL q

2

H4 p-qL q

- r2 + r4 r2

q2 ^2 +

2p 2 p r2 -

:8q Ø 0<, :q Ø

2p

qr -2pr

2 p r2 + >, :q Ø

2p

^ 2, qF

- r2 + r4 r2

Printed by Mathematica for Students

>>


Pythagorean Theorem Roots by Parker Emmerson © 2009-2010.nb

2p

4 p HrL2 q SolveB

2p

H4 p - qL q H4 p - qL q

2 p r2 ::q Ø

:q Ø -

^ 2 ==

4p 3

2

H4 p-qL q

q2

qr-2p ^2 +

2p

- r2 + r4

2 p r2 + >, :q Ø

r2

H4 p-qL q

- r2 + r4 >,

r2

12 p2 r2 - 64 p2 r4

+

1ê3

6 p r2 - 9 r4 - 64 r6 + 3

-

r6 - 13 r8 + 128 r10

3

1ê3

2 p - 9 r4 - 64 r6 + 3

3

r6 - 13 r8 + 128 r10 >,

3 r2

:q Ø -

4p 3

J1 + Â

3 N I12 p2 r2 - 64 p2 r4 M

-

1ê3

12 p r

2

4

6

- 9 r - 64 r + 3

3

6

8

r - 13 r + 128 r

+

10

1ê3

3 N p - 9 r4 - 64 r6 + 3

J1 - Â

r6 - 13 r8 + 128 r10

3

>,

3 r2

:q Ø -

4p 3

J1 - Â

3 N I12 p2 r2 - 64 p2 r4 M

-

1ê3

12 p r

2

4

6

- 9 r - 64 r + 3

3

6

8

r - 13 r + 128 r

10

1ê3

J1 + Â

3 N p - 9 r4 - 64 r6 + 3

3

r6 - 13 r8 + 128 r10 >>

3 r2

Printed by Mathematica for Students

+

2p

H4 p-qL q H4 p-qL q

2p

^ 2, qF

13


14

Pythagorean Theorem Roots by Parker Emmerson © 2009-2010.nb

RevolutionPlot3DB-

4p 3

12 p2 r2 - 64 p2 r4

+

1ê3

6pr

2

4

6

- 9 r - 64 r + 3

3

6

8

r - 13 r + 128 r

1ê3

2 p - 9 r4 - 64 r6 + 3

3

r6 - 13 r8 + 128 r10 , 8r, - 1, 1<F

3 r2

Univocal Radius Solutions Continued by Parker Emmerson © 2009 - 2010

r1 =

r^2 -

SolveB

4 p HrL2 q-HrL2 q2 2p

r^2 -

^2

==

4 p HrL2 q - HrL2 q2 2p

2 p r-r q 2p

^2

==

2pr-rq 2p

, qF

88<<

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10

-


Pythagorean Theorem Roots by Parker Emmerson © 2009-2010.nb

SolveB

2p

H4 p - qL q

^2 -

H4 p - qL q

2 p r2 ::q Ø

- r2 + r4

4 p HrL2 q - HrL2 q2 2p 2 p r2 +

>, :q Ø

r2

SolveB

H4 p - qL q

:8q Ø p<, :q Ø

SolveB

- r2 + r4

H4 p - qL q H4 p - qL q

2 p r2 -

q - HrL2 q2 ^2

>, :q Ø

2p

H4 p-qL q H4 p-qL q

2

2p

q-

2 p r2 + >, :q Ø

2pr-rq 2p

, qF

>>

H4 p-qL q H4 p-qL q

2

q2 ^2

2p

- r2 + r4

==

- r2 + r4 r2

^2 -

r2

, qF

2

2 p r2 +

r2

:8q Ø 2 p<, :q Ø

H4 p-qL q

2p

4p 2p

2p

>>

H4 p-qL q

^2 -

H4 p - qL q

2 p r2 -

2p

2pr-rq

==

- r2 + r4 r2

4p 2p

^2

- r2 + r4 r2

Printed by Mathematica for Students

>>

==

2pr-rq 2p

, qF

15


16

Pythagorean Theorem Roots by Parker Emmerson © 2009-2010.nb

SolveB

2p

4p 2p

H4 p - qL q H4 p - qL q

2 p r2 -

q-

^2 -

2 p r2 + >, :q Ø

r2

2p

H4 p-qL q

2

q2

H4 p-qL q

2p ^2

2p

- r2 + r4

::q Ø

2

H4 p-qL q H4 p-qL q

2p

H4 p-qL q H4 p-qL q

==

-rq

2p

, qF

- r2 + r4 >,

r2

1ê3

p - 9 r4 +

p

:q Ø 2

1ê3

3

1ê3

4

-9 r +

6

r + 27 r

3

-

r6 + 27 r8

3

>,

32ê3 r2

8

1ê3

J1 + Â

J1 - Â

3 Np

:q Ø -

1ê3

3

1ê3

4

-9 r +

6

r + 27 r

3

3 N p - 9 r4 +

+

r6 + 27 r8

3

>,

32ê3 r2

8

1ê3

J1 - Â

J1 + Â

3 Np

:q Ø -

1ê3

3

1ê3

4

-9 r +

6

r + 27 r

3

SolveB

2p

2p

H4 p - qL q H4 p - qL q

H4 p-qL q

-

H4 p-qL q

2p

+

2p

H4 p-qL q

H4 p-qL q

2

2p

q-

H4 p-qL q

^2 -

H4 p-qL q

r6 + 27 r8

3

>>

32ê3 r2

8

4p 2p

3 N p - 9 r4 +

H4 p-qL q

2

H4 p-qL q

q2 ^2

2p

==

q

2p

, qF

88<<

4p SolveB

H4 p-qL q H4 p-qL q

r^2 -

2 p r2 ::q Ø

2p

2

q-

2p

H4 p-qL q H4 p-qL q

2

q2 ^2

2p

- r2 + r4 r2

2 p r2 + >, :q Ø

2p

- r2 + r4 r2

>>

Printed by Mathematica for Students

==

2p

H4 p-qL q H4 p-qL q

-

2p

2p

H4 p-qL q H4 p-qL q

q , qF


Pythagorean Theorem Roots by Parker Emmerson © 2009-2010.nb

4 p HrL2 q SolveB

r^2 -

:8q Ø p<, :q Ø

q2

H4 p-qL q

2 p r2 +

4 p HrL2 q - HrL2 q2

r^2 -

2p - r2 + r4

:8q Ø 2 p<, :q Ø

r^2 -

2p

2 p r2 :8q Ø 0<, :q Ø

H4 p-qL q

2p

H4 p-qL q

, qF

q

H4 p-qL q

, qF

2p

2 p r2 +

q

>>

r2

- r2 + r4

>, :q Ø

r2

H4 p-qL q H4 p-qL q

>,

r2

1ê3

p 9 r4 +

p

:q Ø 2 2 p +

1ê3

3

1ê3

4

9r +

6

r + 27 r

3

-

r6 + 27 r8

3

>,

32ê3 r2

8

1ê3

J1 + Â

J1 - Â

3 Np

:q Ø 4 p -

1ê3

3

1ê3

4

9r +

6

r + 27 r

3

3 N p 9 r4 +

+

3

r6 + 27 r8 >,

32ê3 r2

8

1ê3

J1 - Â

J1 + Â

3 Np

:q Ø 4 p -

1ê3

3

SolveB

1ê3

r^2 -

4

9r +

3

6

r + 27 r

+

^2

==

3

32ê3 r2

8

4 p HrL2 q - HrL2 q2 2p

3 N p 9 r4 +

2pr-rq 2p

, qF

88<<

Printed by Mathematica for Students

q , qF

- r2 + r4

==

- r2 + r4

2p

2p

2pr^2

-

H4 p-qL q

>, :q Ø

4 p HrL2 q - HrL2 q2

H4 p-qL q H4 p-qL q

>>

2 p r2 +

r2

2p

- r2 + r4

2p

==

-

2p

r2 2p

^2

H4 p-qL q H4 p-qL q

==

>, :q Ø

r2

2p

2p ^2

- r2 + r4

2 p r2 -

SolveB

2

H4 p-qL q

2p

2 p r2 -

SolveB

2p

r6 + 27 r8 >>

17


18

Pythagorean Theorem Roots by Parker Emmerson © 2009-2010.nb

4p SolveB

2p

H4 p-qL q H4 p-qL q

r^2 -

2p

q-

H4 p-qL q

q2

H4 p-qL q

^2

2p

==

2pr-rq 2p

, qF

>>

r

4 p HrL2 q SolveB

2

- 1 + r2

2p r::q Ø

2

2p

H4 p - qL q H4 p - qL q

^2 -

2p

H4 p-qL q

2

H4 p-qL q

q2 ^2

2p

2 p r2 :8q Ø - 2 Â p<, 8q Ø 2 Â p<, :q Ø

- r2 + r4

==

2 p r2 + >, :q Ø

r2

2pr-rq 2p

- r2 + r4 >>

r2

ü New Solution 1

2p

4 p HrL2 q SolveB

r^2 -

2 p -2 p q ::r Ø

2

2p ^2

2p

H4 p - qL q -

q2

2p

==

H4 p - qL q +

H4 p-qL q H4 p-qL q

2p

64 p5 q - 80 p4 q2 + 20 p2 q4 - q6

16 p3 q - 20 p2 q2 + q4 2 p -2 p q

:r Ø

H4 p-qL q H4 p-qL q

>,

64 p5 q - 80 p4 q2 + 20 p2 q4 - q6

16 p3 q - 20 p2 q2 + q4

Printed by Mathematica for Students

>>

, qF

-rq , rF


Pythagorean Theorem Roots by Parker Emmerson Š 2009-2010.nb

2 p -2 p q RevolutionPlot3DB

H4 p - qL q +

64 p5 q - 80 p4 q2 + 20 p2 q4 - q6

16 p3 q - 20 p2 q2 + q4

Printed by Mathematica for Students

, 8q, - 2 p, 2 p<F

19


20

Pythagorean Theorem Roots by Parker Emmerson © 2009-2010.nb

ü Solution 1 Substitutions SphericalPlot3DB

2 p -2 p q

1 16 p3 q - 20 p2 q2 + q4

H4 p - qL q +

64 p5 2 p +

p2 - p2 Sin@bD2

- 80 p4 q2 + 20 p2 q4 - q6

8q, - 2 p, 2 p<, 8b, - p ê 2, p ê 2<F

Printed by Mathematica for Students

,


Pythagorean Theorem Roots by Parker Emmerson © 2009-2010.nb

1

SphericalPlot3DB

2

16 p3 q - 20 p2 2 p + 2 p -2 p q

H4 p - qL q +

p2 - p2 Sin@bD2

64 p5 q - 80 p4 q2 + 20 p2 q4 - q6

+ q4 , 8q, - 2 p, 2 p<, 8b, - p ê 2, p ê 2<F

Printed by Mathematica for Students

21


22

Pythagorean Theorem Roots by Parker Emmerson © 2009-2010.nb

1

SphericalPlot3DB

2

16 p3 q - 20 p2 2 p +

p2 - p2 Sin@bD2

+ HqL4

4

2 p -2 p q

H4 p - qL q +

64 p5 q - 80 p4 q2 + 20 p2 2 p +

8q, - 2 p, 2 p<, 8b, - p ê 2, p ê 2<F

Printed by Mathematica for Students

p2 - p2 Sin@bD2

- q6

,


Pythagorean Theorem Roots by Parker Emmerson © 2009-2010.nb

1

SphericalPlot3DB 16 p3 2 p + . 64 p5 2 p +

p2 - p2 Sin@bD2

2 p -2 p q

4

p2 - p2 Sin@bD2

- 80 p4 q2 + 20 p2 2 p +

p2 - p2 Sin@bD2

6

2 p+

H4 p - qL q +

- 20 p2 HqL2 + HqL4

p2 - p2 Sin@bD2

, 8q, - 2 p, 2 p<, 8b, - p ê 2, p ê 2<F

Printed by Mathematica for Students

-

23


24

Pythagorean Theorem Roots by Parker Emmerson © 2009-2010.nb

1

SphericalPlot3DB

2

16 p3 q - 20 p2 2 p + 2 p -2 p q

H4 p - qL q +

p2 - p2 Sin@bD2

64 p5 q - 80 p4 q2 + 20 p2 q4 - q6

4

+ 2 p+

p2 - p2 Sin@bD2

, 8q, - 2 p, 2 p<, 8b, - p ê 2, p ê 2<F

Printed by Mathematica for Students


Pythagorean Theorem Roots by Parker Emmerson © 2009-2010.nb

1

SphericalPlot3DB

2

16 p3 q - 20 p2 2 p +

2 p -2 p q

4p- 2 p+

p2 - p2 Sin@bD2

p2 - p2 Sin@bD2

q +

4

+ 2 p+

p2 - p2 Sin@bD2

64 p5 q - 80 p4 q2 + 20 p2 q4 - q6

8q, - 2 p, 2 p<, 8b, - p ê 2, p ê 2<F

Printed by Mathematica for Students

,

25


26

Pythagorean Theorem Roots by Parker Emmerson © 2009-2010.nb

SphericalPlot3DB 2 p -2 p 2 p +

1 16 p3 q - 20 p2 q2 + q4 p2 - p2 Sin@bD2

H4 p - qL q +

64 p5 q - 80 p4 q2 + 20 p2 q4 - q6

8q, - 2 p, 2 p<, 8b, - p ê 2, p ê 2<F

Printed by Mathematica for Students

,


Pythagorean Theorem Roots by Parker Emmerson © 2009-2010.nb

1

SphericalPlot3DB

4

16 p3 q - 20 p2 q2 + 2 p + -2 p 2 p +

p2 - p2 Sin@bD2

p2 - p2 Sin@bD

2p

2

64 p5 q - 80 p4 q2 + 20 p2 q4 - q6

H4 p - qL q +

8q, - 2 p, 2 p<, 8b, - p ê 2, p ê 2<F

ü New Solution 2

SolveB

::r Ø -

4 p HrL2 q - HrL2 q2

r^2 -

p

2p

1 4p-q

+

1 q

>, :r Ø

p

2p ^2

==

1 4p-q

2p

H4 p-qL q H4 p-qL q

-

2p

+

1 q

>>

Printed by Mathematica for Students

2p

H4 p-qL q H4 p-qL q

q , rF

,

27


28

Pythagorean Theorem Roots by Parker Emmerson © 2009-2010.nb

p

RevolutionPlot3DB

1 4p-q

+

1 q

, 8q, - 2 p, 2 p<F

ü Solution 2 Substitutions

SphericalPlot3DB

1

p 4p-2 p+

p2 - p2 Sin@bD2

+

1 q

, 8b, - p ê 2, p ê 2<, 8q, - 2 p, 2 p<F

Printed by Mathematica for Students


Pythagorean Theorem Roots by Parker Emmerson © 2009-2010.nb

SphericalPlot3DB

1

p 4p-2 p+

SphericalPlot3DB

p

1 4p-q

+

p2 - p2 Sin@bD2

1

+ 2 p+

1 q

, 8q, - 2 p, 2 p<, 8b, - p ê 2, p ê 2<F

, 8b, - p ê 2, p ê 2<, 8q, - 2 p, 2 p<F

p2 - p2 Sin@bD2

Printed by Mathematica for Students

29


30

Pythagorean Theorem Roots by Parker Emmerson © 2009-2010.nb

1

p

SphericalPlot3DB

4p-q

1

+ 2 p+

, 8q, - 2 p, 2 p<, 8b, - p ê 2, p ê 2<F

p2 - p2 Sin@bD2

ü New Solution 3

2p

SolveB

H4 p - qL q H4 p - qL q

-

6 p2 H4 p-qL2

-

p 4 p-q

+

4 p HrL2 q - HrL2 q2

^2 -

2 p2 q2

-

2p

p q

+

4p

H4 p-qL q

+

H4 p-qL2

2p ==

^2

H4 p-qL q

+

4 p-q

2p

H4 p-qL q H4 p-qL q

2p

H4 p-qL q q

::r Ø -

>, 2 -

6 p2 H4 p-qL2

-

p 4 p-q

+

2 p2 q2

-

p q

+

4p

H4 p-qL q H4 p-qL2

+

H4 p-qL q 4 p-q

+

H4 p-qL q

:r Ø

q

>> 2

Printed by Mathematica for Students

q , rF


Pythagorean Theorem Roots by Parker Emmerson Š 2009-2010.nb

-

6 p2 H4 p-qL2

-

p 4 p-q

+

2 p2 q2

-

p q

+

4p

H4 p-qL q H4 p-qL2

+

H4 p-qL q 4 p-q

+

H4 p-qL q q

, 8q, - 2 p, 2 p<F

RevolutionPlot3DB 2

Printed by Mathematica for Students

31


32

Pythagorean Theorem Roots by Parker Emmerson © 2009-2010.nb

SphericalPlot3DB

1 2

6 p2

-

2

4p-2 p+

H4 p - qL q q

2

2

p - p Sin@bD

2

-

p 4p-q

+

2 p2 q

2

-

, 8q, - 2 p, 2 p<, 8b, - p ê 2, p ê 2<F

Theta Solution Representing "Eight Fold Path"

Printed by Mathematica for Students

p q

+

4p

H4 p - qL q H4 p - qL

2

+

H4 p - qL q 4p-q

+


Pythagorean Theorem Roots by Parker Emmerson © 2009-2010.nb

SolveB

2p

H4 p - qL q H4 p - qL q

^2 -

4 p HrL2 q - HrL2 q2 2p

2p ^2

==

2p

H4 p-qL q H4 p-qL q

q

2p

, qF

99q Ø RootA 256 p8 - 512 p7 Ò1 + I256 p6 - 512 p6 r2 M Ò12 + I- 256 p5 + 768 p5 r2 M Ò13 + I64 p4 - 288 p4 r2 + 256 p4 r4 M Ò14 + I32 p3 r2 - 256 p3 r4 M Ò15 + 96 p2 r4 Ò16 - 16 p r4 Ò17 + r4 Ò18 &, 1E=, 9q Ø RootA256 p8 - 512 p7 Ò1 + I256 p6 - 512 p6 r2 M Ò12 + I- 256 p5 + 768 p5 r2 M Ò13 + I64 p4 - 288 p4 r2 + 256 p4 r4 M Ò14 + I32 p3 r2 - 256 p3 r4 M Ò15 + 96 p2 r4 Ò16 - 16 p r4 Ò17 + r4 Ò18 &, 2E=, 9q Ø RootA256 p8 - 512 p7 Ò1 + I256 p6 - 512 p6 r2 M Ò12 + I- 256 p5 + 768 p5 r2 M Ò13 + I64 p4 - 288 p4 r2 + 256 p4 r4 M Ò14 + I32 p3 r2 - 256 p3 r4 M Ò15 + 96 p2 r4 Ò16 - 16 p r4 Ò17 + r4 Ò18 &, 3E=, 9q Ø RootA256 p8 - 512 p7 Ò1 + I256 p6 - 512 p6 r2 M Ò12 + I- 256 p5 + 768 p5 r2 M Ò13 + I64 p4 - 288 p4 r2 + 256 p4 r4 M Ò14 + I32 p3 r2 - 256 p3 r4 M Ò15 + 96 p2 r4 Ò16 - 16 p r4 Ò17 + r4 Ò18 &, 4E=, 9q Ø RootA256 p8 - 512 p7 Ò1 + I256 p6 - 512 p6 r2 M Ò12 + I- 256 p5 + 768 p5 r2 M Ò13 + I64 p4 - 288 p4 r2 + 256 p4 r4 M Ò14 + I32 p3 r2 - 256 p3 r4 M Ò15 + 96 p2 r4 Ò16 - 16 p r4 Ò17 + r4 Ò18 &, 5E=, 9q Ø RootA256 p8 - 512 p7 Ò1 + I256 p6 - 512 p6 r2 M Ò12 + I- 256 p5 + 768 p5 r2 M Ò13 + I64 p4 - 288 p4 r2 + 256 p4 r4 M Ò14 + I32 p3 r2 - 256 p3 r4 M Ò15 + 96 p2 r4 Ò16 - 16 p r4 Ò17 + r4 Ò18 &, 6E=, 9q Ø RootA256 p8 - 512 p7 Ò1 + I256 p6 - 512 p6 r2 M Ò12 + I- 256 p5 + 768 p5 r2 M Ò13 + I64 p4 - 288 p4 r2 + 256 p4 r4 M Ò14 + I32 p3 r2 - 256 p3 r4 M Ò15 + 96 p2 r4 Ò16 - 16 p r4 Ò17 + r4 Ò18 &, 7E=, 9q Ø RootA256 p8 - 512 p7 Ò1 + I256 p6 - 512 p6 r2 M Ò12 + I- 256 p5 + 768 p5 r2 M Ò13 + I64 p4 - 288 p4 r2 + 256 p4 r4 M Ò14 + I32 p3 r2 - 256 p3 r4 M Ò15 + 96 p2 r4 Ò16 - 16 p r4 Ò17 + r4 Ò18 &, 8E== PlotAReARootA 256 p8 - 512 p7 Ò1 + I256 p6 - 512 p6 r2 M Ò12 + I- 256 p5 + 768 p5 r2 M Ò13 + I64 p4 - 288 p4 r2 + 256 p4 r4 M Ò14 + I32 p3 r2 - 256 p3 r4 M Ò15 + 96 p2 r4 Ò16 - 16 p r4 Ò17 + r4 Ò18 &, 8EE, 8r, - 1, 1<E 80 70 60 50 40 30

-1.0

-0.5

0.5

1.0

8 Roots symbolize the eight - fold path.

Printed by Mathematica for Students

33


34

Pythagorean Theorem Roots by Parker Emmerson © 2009-2010.nb

Parker' s Commentary on Fermat' s Last Theorem

Please see The Geometric Pattern of Perception to reference the preliminary investigations. The height of the cone can be calculated in terms of only r and q, thus q is a function of b alone.

Lemma 5

Proof. Since we have shown that q r = 2 p r - 2 p r

1

and r1 Ø

r2 - h2 , we can substitute the expression for r1 , calculated

from the Pythagorean theorem in terms of the height of the cone and the initial radius of the circle, into the expression for q r in terms of the change in circumference of the initial circle to the circle that is the base of the cone into which the circle was transformed. qr=2pr-2p 2 p r Sin@bD

Hr ^ 2 - h ^ 2L , thus, h =

4 p r2 q-r2 q2 2p

= (r Sin[b]). From

4 p q-q2

. So we solve the equation,

4 p q-q2

SolveBr ==

2 p r Sin@bD 4pq-q

::q Ø 2 p -

1=

, qF

2

p2 - p2 Sin@bD2

>, :q Ø 2 p +

p2 - p2 Sin@bD2

2 p Sin@bD 4 p q - q2

Printed by Mathematica for Students

2ph

>>

= r, we note that: r =


Pythagorean Theorem Roots by Parker Emmerson © 2009-2010.nb

The Meaning of n > 2 is n > rq p Hr-r1 L rq J p Hr-r

1L

p Hr - r1 L

2 p Sin@bD

;n>

4pq-q rq p Hr-r1 L

p Sin@bD + rq N p Hr-r1 L

+

rq

rq J p Hr-r

p q - q2

1L

+

+

2 p Sin@bD

2

4pq-q

>

2

p Sin@bD rq N p Hr-r1 L

p q - q2

x ^ n + y ^ n ã z ^ n, where n > 2 q r = 2 p r - 2 p r1 What is 2 ? Solve@q r == ϱ p r - ϱ p r1 , ϱD ::ϱ Ø 2 ==

rq p Hr - r1 L

>>

rq p Hr - r1 L

x ^ n + y ^ n ã z ^ n, where n >

rq p Hr - r1 L rq

x ^ n + y ^ n ã z ^ n, where n > :

2 p r-r q N 2p

p Jr rq

SolveB

p Jr -

2 p r-r q N 2p

==

2 p Sin@bD 4pq-q

+

=

2 p Sin@bD 4pq-q

2 p Sin@bD

2

4pq-q

2 p Sin@bD

+

2

4pq-q

>

2

, rF

2

8< rq

SolveB p Kr -

::r Ø - 2

2 p Sin@bD

==

Hr ^ 2 - h ^ 2L O

4pq-q

2 / 2 p4 h2 Sin@bD2 +

p4 h2

p4 h2

p4 h2

H4 p - qL q Sin@bD3 q

2

4pq-q

, rF

2

4p-q

H4 p - qL q Sin@bD3

2 / 2 p4 h2 Sin@bD2 +

2 p Sin@bD

H4 p - qL q Sin@bD3

p4 h2

ì

q

:r Ø 2

+

H4 p - qL q Sin@bD3

4 p5 h2

q2

- 4 p q3 + q4 + 64 p4 Sin@bD2

H4 p - qL q Sin@bD3 4p-q

ì

+

+

4 p5 h2

H4 p - qL q Sin@bD3

- 4 p q3 + q4 + 64 p4 Sin@bD2

Printed by Mathematica for Students

>,

q2

>>

+

+

35


36

Pythagorean Theorem Roots by Parker Emmerson © 2009-2010.nb

2 . 2 p4 h2 Sin@bD2 +

ContourPlot3DB:- 2 4 p 5 h2

H4 p - qL q Sin@bD3 q2

- 4 p q3 + q4 + 64 p4 Sin@bD2 4 p 5 h2

, 2

H4 p - qL q Sin@bD3 q2

- 4 p q3 + q4 + 64 p4 Sin@bD2

2p

H4 p-qL q H4 p-qL q

SolveB p

2p

::q Ø 2 p -

H4 p-qL q H4 p-qL q

-

2p

2p

p2 - p2 Sin@bD2

+

+

p 4 h2

4p-q

p 4 h2

H4 p - qL q Sin@bD3

2 . 2 p4 h2 Sin@bD2 + p 4 h2

+

ì

q

H4 p - qL q Sin@bD3

p 4 h2

4p-q

H4 p - qL q Sin@bD3 q

ì

>, 8h, - 1, 1<, 8q, - 2 p, 2 p<, 8b, - p ê 2, p ê 2<F

q ==

H4 p-qL q H4 p-qL q

H4 p - qL q Sin@bD3

-

2p

H4 p-qL q H4 p-qL q

q

2 p Sin@bD 4pq-q

+

2

4pq-q

2p

>, :q Ø 2 p +

p2 - p2 Sin@bD2

Printed by Mathematica for Students

2 p Sin@bD

>>

2

, qF

+


Pythagorean Theorem Roots by Parker Emmerson © 2009-2010.nb

2p

H4 p-qL q H4 p-qL q

SolveB 2p

p

H4 p-qL q H4 p-qL q

-

2p

2p

q

2 p Sin@bD

== H4 p-qL q

H4 p-qL q

-

2p

H4 p-qL q H4 p-qL q

2 p Sin@bD

+

4 p q - q2

q

4 p q - q2

2p

H4 p - qL q ::b Ø ArcSinB

F>>

2p rq

SolveB p r-

==

4 p r2 q-r2 q2

r^2 -

2p

2 p Sin@bD 4pq-q

+

2 p Sin@bD

2

4pq-q

, rF

2

^2

8< rq

SolveB p r-

::q Ø 2 p -

r^2 -

==

4 p r2 q-r2 q2 2p

p2 - p2 Sin@bD2

2 p Sin@bD

+

2 p Sin@bD

4 p q - q2 ^2

>, :q Ø 2 p +

p2 - p2 Sin@bD2

>,

p2 Sin@bD2

:q Ø 2 31ê3 - 9 p3 Sin@bD2 +

- 9 p3 Sin@bD2 +

3

1ê3

27 p6 Sin@bD4 + p6 Sin@bD6

3

27 p6 Sin@bD4 + p6 Sin@bD6

>,

3 N p2 Sin@bD2

:q Ø 31ê3 - 9 p3 Sin@bD2 +

J1 - Â

3

3 N - 9 p3 Sin@bD2 +

-

1ê3

32ê3

J1 + Â

27 p6 Sin@bD4 + p6 Sin@bD6

1ê3

+

1ê3

27 p6 Sin@bD4 + p6 Sin@bD6

3

>,

32ê3 J1 - Â

3 N p2 Sin@bD2

:q Ø 3

1ê3

J1 + Â

3

2

- 9 p Sin@bD +

3

3 N - 9 p3 Sin@bD2 +

, qF

4 p q - q2

6

4

6

27 p Sin@bD + p Sin@bD

6

1ê3

27 p6 Sin@bD4 + p6 Sin@bD6

3 32ê3

Printed by Mathematica for Students

+

1ê3

>>

, bF

37


38

Pythagorean Theorem Roots by Parker Emmerson © 2009-2010.nb

p2 Sin@bD2

RevolutionPlot3DB2

1ê3

3

1ê3

2

3

- 9 p Sin@bD +

4

6

6

27 p Sin@bD + p Sin@bD

3

-

6

1ê3

- 9 p3 Sin@bD2 +

27 p6 Sin@bD4 + p6 Sin@bD6

3

, 8b, - p ê 2, p ê 2<F

32ê3

rq

SolveB p r-

::b Ø ArcSinB

r^2 -

== 2

2

4 p r q-r q

rq 2p 2pr-

2

2p

2 p Sin@bD 4 p q - q2

+

2 p Sin@bD

, bF

4 p q - q2

^2

H4 p - qL q F>> 2

r H2 p - qL

RevolutionPlot3DBArcSinB

2

rq 2p 2pr-

H4 p - qL q 2

r H2 p - qL

F, 8r, - 1, 1<, 8q, - 2 p, 2 p<F 2

For n = 3,

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Pythagorean Theorem Roots by Parker Emmerson © 2009-2010.nb

rq

SolveBx ^

p Jr -

2 p r-r q 2p

rq

z^

2 p r-r q

p Jr -

2p

+

+

2 p Sin@bD

N

p Jr -

4 p q - q2

2 p Sin@bD

N

rq

+ y^

4pq-q

2 p r-r q 2p

+

2 p Sin@bD

N

4 p q - q2

, xF

2 1

rq

::x Ø - y

r :=

p r-

2p

2 p r-r q

+

2 p Sin@bD H4 p-qL q

2p

rq

+z

p r-

2 p r-r q 2p

+

2 p Sin@bD H4 p-qL q

2+

2 p Sin@bD H4 p-qL q

>>

H4 p - qL q H4 p - qL q

q := 2 p +

p2 - p2 Sin@bD2 1 rq

ContourPlot3DB - y

p r-

2 p r-r q 2p

+

2 p Sin@bD H4 p-qL q

rq

+z

p r-

2 p r-r q 2p

+

2 p Sin@bD 2+ H4 p-qL q

2 p Sin@bD H4 p-qL q

8y, - 5, 5<, 8z, - 5, 5<, 8b, - p ê 2, p ê 2<F

b := ArcSinB

H4 p - qL q 2p

F

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,

ã

39


40

Pythagorean Theorem Roots by Parker Emmerson Š 2009-2010.nb

1 rq

ContourPlot3DB - y

p r-

2 p r-r q 2p

+

2 p Sin@bD H4 p-qL q

rq

+z

p r-

2 p r-r q 2p

+

2 p Sin@bD 2+ H4 p-qL q

2 p Sin@bD H4 p-qL q

8y, - 5, 5<, 8z, - 5, 5<, 8q, - 2 p, 2 p<F

Printed by Mathematica for Students

,


Pythagorean Theorem Roots by Parker Emmerson © 2009-2010.nb

rq

SolveBx ^

2 p r-r q

p Jr -

2p

rq

z^

p Jr -

2 p r-r q 2p

+

+

2 p Sin@bD

N

p Jr -

4 p q - q2

2 p Sin@bD

N

rq

+ y^

4pq-q

2 p r-r q 2p

+ N

, yF

2 1

rq

::y Ø - x

p r-

SolveBr ==

2p

::q Ø

2 p r2 +

2 p Sin@bD H4 p-qL q

2p

rq

+z

H4 p - qL q H4 p - qL q

2 p r2 -

q :=

+

2 p r-r q

p r-

2 p r-r q 2p

2 p Sin@bD H4 p-qL q

2+

2 p Sin@bD H4 p-qL q

>>

, qF

- r2 + r4 r2

+

2 p r2 + >, :q Ø

- r2 + r4 r2

>>

- r2 + r4 r2

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2 p Sin@bD 4 p q - q2

ã

41


42

Pythagorean Theorem Roots by Parker Emmerson Š 2009-2010.nb

1 rq

ContourPlot3DB - x

p r-

2 p r-r q 2p

+

2 p Sin@bD H4 p-qL q

rq

+z

p r-

2 p r-r q 2p

+

2 p Sin@bD 2+ H4 p-qL q

2 p Sin@bD H4 p-qL q

8x, - 1, 1<, 8z, - 1, 1<, 8r, - 2 p, 2 p<F

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,


Pythagorean Theorem Roots by Parker Emmerson © 2009-2010.nb

1 rq

ContourPlot3DB - x

p r-

2 p r-r q 2p

rq

SolveBx ^

p Jr -

2 p r-r q 2p

rq

z^

p Jr -

2 p r-r q 2p

+ N

+ N

+

2 p Sin@bD H4 p-qL q

rq

+z

p r-

2 p Sin@bD 4pq-q

2 p Sin@bD 4pq-q

2 p r-r q

2 p Sin@bD

+

2+ H4 p-qL q

2p

H4 p-qL q

rq

+ y^

p Jr -

2

2 p Sin@bD

2 p r-r q 2p

+ N

, zF

2 1

rq

::z Ø x

p r-

2 p r-r q 2p

+

2 p Sin@bD H4 p-qL q

rq

+y

p r-

2 p r-r q 2p

+

2 p Sin@bD H4 p-qL q

2+

2 p Sin@bD H4 p-qL q

>>

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, 8x, - 1, 1<, 8z, - 1, 1<, 8r, - 5, 5<F

2 p Sin@bD 4pq-q

2

ã

43


44

Pythagorean Theorem Roots by Parker Emmerson © 2009-2010.nb

Form and Formality ("Gibson - Stillwell" - Emmerson Formulation of Fermat's Last Theorem) It is important to note that it is an area of the retina that is affected by radiant energy, not just a single point. Gibson’s reason for believing that the geometry of transformations is important to visual perception is that, “transformations are usually represented on a plane, however, whereas the retinal image is a projection on a curved surface,” (The Visual World, 153) (Gibson, James, J.. The Perception of the Visual World. Cambridge, Mass.: The Riverside Press, 1950. Print. (All further references to this source will be cited parenthetically in the text).), and that, “the actual retinal image on a curved surface is related to the hypothetical image on a picture-plane only by such a non-rigid transformation” (The Visual World, 153). A transformation in geometry is the mapping of one point onto another. Isometries, “are defined as the transformations that preserve distance” (The Four Pillars of Geometry, 145) (Stillwell, John. The Four Pillars of Geometry (Undergraduate Texts in Mathematics). 1 ed. New York: Springer, 2005. Print.). In essence, the distance of the initial radius is preserved through the transformation of a circle into a cone so long as the height is orthogonal to the base of the cone and the initial radius is always the slant of the cone. Next, we see the diagram to which Gibson was referring when considering the notion of a transformation onto a picture plane.

(The Visual World, 79).

In being preserved, the initial radius is considered an invariant. Stillwell comments about the picture plane that, “the line from (-1, 1) to (n, 0) crosses the y-axis at y = n/(n+1)” (The Four Pillars of Geometry, 91). This supposes that the eye is approximated like a point and that it is at the position of (-1, 1) in the Cartesian coordinate system. In the “coordinate system” described by The Geometric Pattern of Perception Theorems (Emmerson, 2009), the y-axis in general is described by the height of a cone. In relation to this diagram, in terms of the y intercept, the height of the cone would be changing with respect to both the initial radius (slant of the cone) and the angle taken out of the initial circle (the angle made between the line from the eye to the x-axis changes is a function of solely the angular amount taken out of the initial circle). Further mathematical analysis of optical infinity with relation to the horizon line and geometric system is needed, but perceived difference in circumferences as an arc length will be a useful formula. Gibson says that, “only because light is structured by the substantial environment can it contain information about it” (Ecological Approach, 86). The basic equation for an arc length as a difference in circumferences describes an even surface layout. Thus, for even surfaces, the equation that delivers that surface may be used as a linguistic device (in combination with rotation, or specifying the “adumbration” of the viewed surface) for describing the structuring of the light in the environment relevant to the perception of even surface layout. The expression for “phenomenal velocity” tells us “how” motion in general is essentially structured, thus this includes the motion of light. However, this still needs specific interpretation.

From Stillwell ' s statement, y = n ê Hn + 1L, h =

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4 p r2 q - r2 q2 2p

= n ê Hn + 1L

(1)


Pythagorean Theorem Roots by Parker Emmerson © 2009-2010.nb

SolveB

4 p r2 q - r2 q2 2p

== n ê Hn + 1L, nF

r2 H4 p - qL q

::n Ø

45

>>

(2)

2

2p-

r H4 p - qL q

x^n + y^n = z^n =

2pr-rq 2p

^n +

4 p r2 q - r2 q2 2p

(3) ^ n ã r ^ n == Indeterminate ã Indeterminate

Proof !?!? Indeterminate indeed equals an indeterminate. However, there are still other configurations that may lead to trying to solve for a complex infinity. Complex infinity has specific geometric meaning-expression available in The Geometric Pattern of Perception (Emmerson, 2009). Now the question is, "where does the indeterminate equal the indeterminate?"

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On Formulations of the Pythagorean Theorem