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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