Thursday, January 24, 2008

Gauss: Constructibility of Heptadecagon

In the previous blog, I showed an algorithm for constructing the heptadecagon (a regular 17-sided figure). In today's blog, I will show a proof that based on the Gauss's solution for the seventeenth roots of unity in terms of radicals, the algorithm in the previous blog really works.

Lemma 1:

if α = 2π/17

and

ζk = cos kα + isin k α

then

ζk + ζ17 - k = 2 cos kα

Proof:

(1) Using De Moivre's Formula for roots of unity (see Corollary 1.1, here), we note that:

if α = 2π/17, then:

ζ = cos α + isinα is a seventeenth root of unity.

(2) Using De Moivre's Formula (see Theorem 1, here), we note that:

ζk = (cos α + isin α)k = cos kα + isin k α

ζ17-k = (cos α + isin α)17-k = cos ([17 - k]α) + isin([17 -k]α) = cos ([17 - k]2π/17) + isin([17 - k]2π/17) = cos(2π - kα) + isin(2π - kα) = cos(-kα) + isin(-kα) = cos(kα) - isin(kα)

(3) So that we have:

ζk + ζ17-k = cos kα + isin kα + cos kα - isink α = 2cos kα

QED

Lemma 2:

If C is the least positive acute angle such that tan 4C = 4

with

α = 2π/17

ζk = cos kα + isin k α

and:

x1 = ζ + ζ9 + ζ13 + ζ15 + ζ16 + ζ8 + ζ4 + ζ2 x2 = ζ3 + ζ10 + ζ5 + ζ11 + ζ14 + ζ7 + ζ12 + ζ6

then:

x1 = 2 tan 2C x2 = -2cot 2C

Proof:

(1) Using step #1 thru step #7 in Theorem 1, here, we have:

x1 and x2 are the solutions to:

x2 + x - 4 = 0

(2) Since tan 4C = 4, it follows that cot 4C = 1/(tan 4C) = 1/4

(3) So 4xcot 4C = 4x(1/4) = x
(4) And we have:

x2 + 4xcot 4C - 4 = 0
(5) Using the quadratic equation (see Theorem, here), we have:

x = (1/2)[-4cot4c ± √(4cot 4c)2 + 16]

(6) Using Lemma 1, here, we have:

x = (1/2)[-4(1/2)(cot 2c - tan2c) ± √16(1/4)(cot 2c - tan 2c)2 + 16) ]= = (tan 2c - cot2c) ± (1/2)√4[cot2(2c) - 2cot(2c)tan(2c) + tan2(2c)] + 16 = = (tan 2c - cot 2c) ± √cot2(2c) - 2 + tan2(2c) + 4 = = (tan 2c - cot 2c) ± √cot2(2c) + tan2(2c) + 2 = = (tan 2c - cot 2c) ± √[cot(2c) + tan(2c)]2 = = (tan 2c - cot 2c) ± [cot(2c) + tan(2c)]

(7) Since x1 is greater than x2, we have:

x1 = 2 tan 2c x2 = -2 cot 2c

QED

Lemma 4:

If C is the least positive acute angle such that tan 4C = 4

with:

α = 2π/17

ζk = cos kα + isin k α

and:

y1 = ζ1 + ζ13 + ζ16 + ζ4

y2 = ζ9 + ζ15 + ζ8 + ζ2

y3 = ζ3 + ζ5 + ζ14 + ζ12

y4 = ζ10 + ζ11 + ζ7 + ζ6

then:

y1 = tan(C + π/4) y2 = tan(C - π/4) y3 = tan C y4 = -cot C

Proof:

(1) From steps #9 thru #13 in Theorem 1, here, we have:

y1, y2 are solutions to:

y2 - x1y - 1 = 0

and

y3, y4 are solutions to:

y2 - x2y - 1 = 0

(2) Let's solve first for y1, y2

(3) Using the quadratic equation (see Theorem 1, here), we get:
y = (1/2)(x1 ± √x12 + 4)

(4) From Lemma 2 above, we know that x1 = 2 tan 2c

(5) So that we have:

y = (1/2)(2 tan 2c ± √4(tan 2c)2 + 4 = = tan 2c ± √(tan 2c)2 + 1

(6) Using Lemma 4, here, we have:

tan 2c ± √(tan 2c)2 + 1 = (2 tan c)/(1 - tan2c) ± √(4 tan2 c)/(1 - 2tan2c + tan4 c) + 1 = = (2 tan c)/(1 - tan2c) ± √(4 tan2 c + 1 - 2tan2c + tan4 c )/(1 - 2tan2c + tan4 c) = = (2 tan c)/(1 - tan2c) ± √(2tan2 c + 1 + tan4 c )/(1 - 2tan2c + tan4 c) = = (2 tan c)/(1 - tan2c) ± √(1 + tan2 c )2/(1 - tan2c)2 = = (2 tan c)/(1 - tan2c) ± (1 + tan2c)/(1 - tan2c) = = (2 tan c ± [1 + tan2c])/(1 - tan2c)
(7) This then gives us:

(2 tan c + 1 + tan2c)/(1 - tan2c) = = (tan c + 1)(tan c + 1)/(1 + tan c)(1 - tan c) = (tan c + 1)/(1 - tan c)

and

(2 tan c - 1 - tan2c)/(1 - tan2c) = (tan2c - 2tan c + 1)/(tan2c - 1) = = (tan c - 1)(tan c - 1)/(tan c - 1)(tan c + 1) = (tan c - 1)/(tan c + 1)
(8) Since tan(π/4) = 1 (see Lemma 1, here), we can restate this as:

(1 + tan c)/(1 - tan c) = (tan π/4 + tan c)/(1 - (tan c)(tan π/4)) (tan c - 1)/(tan c + 1) = (tan c - tan π/4)/(1 + (tan c)(tan π/4))

(9) Using Lemma 3, here, we have:

(1 + tan c)/(1 - tan c) = (tan π/4 + tan c)/(1 - (tan c)(tan π/4)) = tan(c + π/4)

(10) Using Corollary 3.1, here, we have:

(tan c - 1)/(tan c + 1) = (tan c - tan π/4)/(1 + (tan c)(tan π/4)) = tan(c - π/4)

(11) Since y1 is greater than y2,

y1 = tan(c + π/4) y2 = tan(c - π/4)

(11) Now, let's solve for y3, y4

(12) Using the quadratic equation (see Theorem 1, here), we get:

y = (1/2)(x2 ± √x22 + 4)

(13) From Lemma 2 above, we know that x2 = -2 cot 2c

(14) So that we have:

y = (1/2)(-2 cot 2c ± √4(cot 2c)2 + 4 = = -cot 2c ± √(cot 2c)2 + 1

(15) Using Lemma 1, here, we have:

-cot 2c ± √(cot 2c)2 + 1 = -(1/2)(cot c - tan c) ± √(1/4)(cot c - tan c)2 + 1 = = (1/2)(tan c - cot c) ± √(1/4)(cot2c + tan2c - 2(cot c)(tan c)) + (4/4) = = (1/2)(tan c - cot c) ± √(1/4)(cot2c + tan2c - 2 + 4) = = (1/2)(tan c - cot c) ± √(1/4)(cotc + tanc)2 = = (1/2)(tan c - cot c) ± (1/2)(tan c + cot c)
(16) Since y3 is greater than y4, this gives us:

y3 = tan c y4 = -cot c

QED
Corollary 4.1:

If C is the least positive acute angle such that tan 4C = 4

with:

α = 2π/17 tan c = 2 cos 3α + 2 cos 5α tan (c - π/4) = 2 cos 2α + 2 cos8α

Proof:

(1) Let us define the following:

y3 = ζ3 + ζ5 + ζ14 + ζ12

and

y2 = ζ9 + ζ15 + ζ8 + ζ2

where

ζk = cos kα + isin k α

(2) Using Lemma 1 above:

y3 =(ζ3 + ζ14) + (ζ5 + ζ12) = 2cos 3α + 2cos 5α

y2 = 8 + ζ9) + 2 + ζ15) = 2 cos 8α + 2 cos 2 α

(3) Using Lemma 5, here, we have:

2 cos(3α) * 2cos(5α) = 2*[2 cos(3α)*cos(5α)] = 2*[cos(5 + 3)α + cos(5 - 3)α] = 2[cos(8α) + cos(2α)] = 2cos(8α) + 2cos(2α)

(4) From Lemma 4 above, we then have:

tan c = 2 cos 3α + 2 cos 5α tan (c - π/4) = 2 cos 2α + 2 cos8α = 2cos(3α)*2cos(5α)
QED

Theorem 5: Constructibility of Heptadecagon using compass and ruler




































Proof:

(1) Let CD, CP0 two radii of a circle C1 that are perpendicular to each other.
(2) Let CG be 1/4 the length of CD (bisector of a bisector).
(3) Let ∠ CGL be 1/4 ∠ CGP0 with L on CP0 (bisector of a bisector).
(4) Let ∠ LGP = π/4 (bisector a right angle since right angle = π/2, see here for review of radians) where P is on CP0.
(5) Let Q be the midpoint of PP0 and draw a circle C2 with center Q and radius QP0.
(6) Let R be the point where the circle C1 intersects with CD.
(7) Draw a circle C3 with center L and radius equal to RL.
(8) Let S,Q be the points where the circle C3 intersects with CP0.
(9) Let SP5, QP3 be lines parallel to CP0 that intersect with circle C1.
(10) Let C = measurement ∠ CGL
(11) Then 4C = measurement ∠ CGP0
(12) cos P0CP3 = CQ/CP3 = CQ/CP0 [See here for definition of cosine]
(13) cos P0CP5 = cos (π/2 + π/2 - SCP5) = cos (π - SCP5)
(14) Using cos(a + b) = (cos a)(cos b) - (sin a)(sin b) (see Theorem 2, here) and cos(-x) = cos(x) (see Property 9, here), we have:

cos(π - SCP5) = (cos π)(cos -SCP5) - (sin π)(sin -SCP5) = (-1)cos(SCP5) - (0)(-SCP5) = -cos(SCP5)

(15) So, cos P0CP5 = -cos(SCP5) = -SC/CP5 = -SC/CP0

(16) This gives us that:

2cos P0CP3 + 2cos P0CP5 = 2(CQ - SC)/CP0

(17) Since CQ = CL + LQ and SC = SL - CL and SL=LQ:

SC = SL - CL = LQ - CL

CQ - SC = (CL + LQ) - (LQ - CL) = CL + LQ - LQ + CL = 2CL

(18) This gives us that:

2(CQ - SC)/CP0 = 2(2CL)/CP0 = 4CL/CP0

(19) Since CG is 1/4 of CD and CD=CP0, we have:

4CL/CP0 = 4CL/CD = 4CL/(4CG) = CL/CG

(20) Since GCL is a right angle and C = measurement ∠ CGL, we have:

tan CGL = tan c = sin CGL/cos CGL = (LC/GL)/(GC/GL) = LC/GC = 2cos P0CP3 + 2cos P0CP5

(21) Since cos P0CP3 =CQ/CP0 and cos P0CP5 = -SC/CP0, we have:

2 cos(P0CP3)*2cos(P0CP5) = -4*(SC)(CQ)/(CP0)(CP0)

(22) Using the Pythagorean Theorem (see Theorem, here):

CR2 = CL2 + RL2 = (CL - RL)(CL + RL)

(23) Since LQ=LR and LS=RL:

CL + RL = CL + LQ = CQ

CL - RL = CL - LS = SC

(24) So we have:

CR2 = (CL - LR)(CL + LR) = (CQ)(SC)

(25) Using the Pythagorean Theorem:

(RC)2 = RQ2 - CQ2 = (RQ - CQ)(RQ + CQ)

(26) Now RQ = QP and RQ=QP0 so that:

(RQ - CQ) = (QP - CQ) = PC

(RQ + CQ) = (QP1 + CQ) = CP1

(27) This gives us that (RC)2 = (PC)(CP0) so that:

-4(RC)2/(CP0)2 = -4(PC)(CP0)/(CP0)(CP0) = -4(PC)/(CP0)

(28) Since CG = (1/4)CD = (1/4)CP0, we have:

-4(PC)/(CP0) = -4(PC)/(4CG) = -PC/CG

(29) Also:

tan ∠ CGP = sin ∠ CGP/cos ∠ CGP = (PC/GP)/(CG/GP) = PC/GC

(30) Since ∠ CGP = ∠ LGP - ∠ CGL, we have:

tan ∠ CGP = tan (∠ LGP - ∠ CGL) = tan(π/4 - C)

and this means:

-PC/GC = -tan(π/4 - C) = tan(C - π/4)

(31) Using Corollary 4.1 above, we have:

2cos ∠ P0CP3 + 2cos ∠ P0CP5 = 2 cos 3α + 2 cos 5α = tan c where α = 2π/17

2 cos(∠P0CP3)*2cos(∠P0CP5) = 2cos(3α)*2cos(5α) = tan (c - π/4)

(32) Now, it is clear that ∠ P0CP3 and ∠ P0CP5 map onto and .

(33) Since ∠ P0CP5 is greater than ∠P0CP3, it is clear that:

∠ P0CP3 = 3α

∠ P0CP5 = 5α

QED

References

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