10 Dimensions: The Wavelength with Constant Value by Parker Emmerson ©2009-2010

8/8/2019 10 Dimensions: The Wavelength with Constant Value by Parker Emmerson ©2009-2010

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XVIII. The Wavelength with Constant Value

In this section, I will make a few considerations on the meanings of derivatives within the system and the proper applica-

tion of them. Because the system described is complete (its parameters are all tied together and dependent on each other in some

way or another), there are certain hypothetical inconsistencies.

(1)2 p r - 2 p r1 = q r

(2)H2 p r - 2 p r1L

q= r

(3)

velocity = wavelength of frequency = v = l ƒ = r ƒ =

r ê t = r H1 ê Hq ê H2 pLLL =2 p r

q=

2 p s

q2

=

2 pH2 p r-2 p r1L

q

q=

2 p2 p r-2 p r

2-h2

q

q=

2 p 2 p r - 2 p r2 - h2

q2

=

2 p 2 p r - 2 p r2

-

4 p r2 q-r2 q2

2 p

2

q2

=

H1 ê H1 ê H2 pLLL DB 4 p r2 q - r2 q2

2 p, qF =

4 p r2 - 2 r2 q

2 4 p r2 q - r2 q2

ü h

SolveB2 p 2 p r - 2 p r2 -

Hh

L2

q2

==4 p r2 - 2 r2 q

2 4 p r2 q - r2 q2

, hF::h Ø -

1

4 p2/ 64 p4 r2 -

256 p5 r2

4 p - q+ 16 p3 r2 q + 4 p2 r2 q2 + r2 q4 +

16 p3r r

2 H4 p - qL q -64 p4 r r2 H4 p - qL q

4 p - q+ 8 p2

r q r2 H4 p - qL q >,

:h Ø1

4 p2/ 64 p4 r2 -

256 p5 r2

4 p - q+ 16 p3 r2 q + 4 p2 r2 q2 + r2 q4 + 16 p3 r r2 H4 p - qL q -

64 p4 r r2 H4 p - qL q

4 p - q+ 8 p

2r q r

2 H4 p - qL q >>

Printed by Mathematica for Students



RevolutionPlot3DB-1

4 p 2/ 64 p

4 r2-

256 p 5 r2

4 p - q+ 16 p

3 r2q + 4 p

2 r2q2

+ r2q4

+ 16 p 3 r r2 H4 p - qL q -

64 p 4 r r2 H4 p - qL q

4 p - q

+ 8 p 2 r q r2

H4 p - q

Lq ,

8r, - 1, 1

<,

8q, - 4 p , 4 p

<F

2 The Wavelength with Constant Value by Parker Emmerson ©2009-2010.nb




ü r

SolveB 2 p 2 p r - 2 p r2 - HhL2q2

==4 p r2 - 2 r2 q

2 4 p r

2

q - r

2

q

2

, rF99r Ø -,I-I256 p

7h2

q3M ë I- 1024 p

7q3

+ 1280 p6

q4

- 512 p5

q5

+ 80 p4

q6

- 32 p3

q7

+ 24 p2

q8

- 8 p q9

+ q10M +I320 p

6h2

q4M ë I- 1024 p

7q3

+ 1280 p6

q4

- 512 p5

q5

+ 80 p4

q6

- 32 p3

q7

+ 24 p2

q8

- 8 p q9

+ q10M -I128 p5 h2 q5M ë I- 1024 p7 q3 + 1280 p6 q4 - 512 p5 q5 + 80 p4 q6 - 32 p3 q7 + 24 p2 q8 - 8 p q9 + q10M +I16 p4 h2 q6M ë I- 1024 p7 q3 + 1280 p6 q4 - 512 p5 q5 + 80 p4 q6 - 32 p3 q7 + 24 p2 q8 - 8 p q9 + q10M -I128 p

6 ,I256 p5

h4

q3

- 448 p4

h4

q4

+ 304 p3

h4

q5

- 100 p2

h4

q6

+ 16 p h4

q7

- h4

q8MM ëI- 1024 p7 q3 + 1280 p6 q4 - 512 p5 q5 + 80 p4 q6 - 32 p3 q7 + 24 p2 q8 - 8 p q9 + q10MM=,9r Ø ,I-I256 p7 h2 q3M ë I- 1024 p7 q3 + 1280 p6 q4 - 512 p5 q5 + 80 p4 q6 - 32 p3 q7 + 24 p2 q8 - 8 p q9 + q10M +I320 p6 h2 q4M ë I- 1024 p7 q3 + 1280 p6 q4 - 512 p5 q5 + 80 p4 q6 - 32 p3 q7 + 24 p2 q8 - 8 p q9 + q10M -I128 p5 h2 q5M ë I- 1024 p7 q3 + 1280 p6 q4 - 512 p5 q5 + 80 p4 q6 - 32 p3 q7 + 24 p2 q8 - 8 p q9 + q10M +

I16 p

4h2

q6

M ë I- 1024 p

7q3

+ 1280 p6

q4

- 512 p5

q5

+ 80 p4

q6

- 32 p3

q7

+ 24 p2

q8

- 8 p q9

+ q10

M-

I128 p6

,I256 p5

h4

q3

- 448 p4

h4

q4

+ 304 p3

h4

q5

- 100 p2

h4

q6

+ 16 p h4

q7

- h4

q8

MM ëI- 1024 p7 q3 + 1280 p6 q4 - 512 p5 q5 + 80 p4 q6 - 32 p3 q7 + 24 p2 q8 - 8 p q9 + q10MM=,9r Ø -,I-I256 p7

h2

q3M ë I- 1024 p

7q3

+ 1280 p6

q4

- 512 p5

q5

+ 80 p4

q6

- 32 p3

q7

+ 24 p2

q8

- 8 p q9

+ q10M +I320 p

6h2

q4M ë I- 1024 p

7q3

+ 1280 p6

q4

- 512 p5

q5

+ 80 p4

q6

- 32 p3

q7

+ 24 p2

q8

- 8 p q9

+ q10M -I128 p5 h2 q5M ë I- 1024 p7 q3 + 1280 p6 q4 - 512 p5 q5 + 80 p4 q6 - 32 p3 q7 + 24 p2 q8 - 8 p q9 + q10M +I16 p4 h2 q6M ë I- 1024 p7 q3 + 1280 p6 q4 - 512 p5 q5 + 80 p4 q6 - 32 p3 q7 + 24 p2 q8 - 8 p q9 + q10M +I128 p

6 ,I256 p5

h4

q3

- 448 p4

h4

q4

+ 304 p3

h4

q5

- 100 p2

h4

q6

+ 16 p h4

q7

- h4

q8MM ëI- 1024 p7 q3 + 1280 p6 q4 - 512 p5 q5 + 80 p4 q6 - 32 p3 q7 + 24 p2 q8 - 8 p q9 + q10MM=,9r Ø ,I-I256 p7 h2 q3M ë I- 1024 p7 q3 + 1280 p6 q4 - 512 p5 q5 + 80 p4 q6 - 32 p3 q7 + 24 p2 q8 - 8 p q9 + q10M +I320 p6 h2 q4M ë I- 1024 p7 q3 + 1280 p6 q4 - 512 p5 q5 + 80 p4 q6 - 32 p3 q7 + 24 p2 q8 - 8 p q9 + q10M -I128 p5 h2 q5M ë I- 1024 p7 q3 + 1280 p6 q4 - 512 p5 q5 + 80 p4 q6 - 32 p3 q7 + 24 p2 q8 - 8 p q9 + q10M +

I16 p4

h2

q6

M ë I- 1024 p7

q3

+ 1280 p6

q4

- 512 p5

q5

+ 80 p4

q6

- 32 p3

q7

+ 24 p2

q8

- 8 p q9

+ q10

M +I128 p6 ,I256 p5 h4 q3 - 448 p4 h4 q4 + 304 p3 h4 q5 - 100 p2 h4 q6 + 16 p h4 q7 - h4 q8MM ëI- 1024 p7 q3 + 1280 p6 q4 - 512 p5 q5 + 80 p4 q6 - 32 p3 q7 + 24 p2 q8 - 8 p q9 + q10MM==

The Wavelength with Constant Value by Parker Emmerson ©2009-2010.nb 3




ü q

SolveB 2 p 2 p r - 2 p r2 - HhL2q2

==4 p r2 - 2 r2 q

2 4 p r

2

q - r

2

q

2

, qF::q Ø RootB- 128 p5 r2 + 64 p5 h2 + 128 p5 r r2 - h2 +

32 p4 r2 - 16 p4 h2 - 32 p4 r r2 - h2 Ò1 + 4 p2 r2 Ò13

- 4 p r2 Ò14

+ r2 Ò15

&, 1F>,

:q Ø RootB- 128 p5 r2 + 64 p5 h2 + 128 p5 r r2 - h2 + 32 p4 r2 - 16 p4 h2 - 32 p4 r r2 - h2 Ò1 +

4 p2

r2

Ò13

- 4 p r2

Ò14

+ r2

Ò15

&, 2F>,


4 p2

r2

Ò13

- 4 p r2

Ò14

+ r2

Ò15

&, 3F>,

:q Ø Root

B- 128 p5 r2 + 64 p5 h2 + 128 p5 r r2 - h2 + 32 p4 r2 - 16 p4 h2 - 32 p4 r r2 - h2 Ò1 +

4 p2

r2

Ò13

- 4 p r2

Ò14

+ r2

Ò15

&, 4F>,


4 p2r2 Ò1

3- 4 p r

2 Ò14

+ r2 Ò1

5&, 5F>>

Plot3DBRootB- 128 p 5 r2 + 64 p 5 h2 + 128 p 5 r r2 - h2 + 32 p 4 r2 - 16 p 4 h2 - 32 p 4 r r2 - h2 Ò1 +

4 p 2 r2 Ò13- 4 p r2 Ò14

+ r2 Ò15 &, 1F, 8r, - 1, 1<, 8h, - 1, 1<F





SolveB 2 p 2 p r - 2 p r2 - Hr Sin@bDL2q2

==4 p r2 - 2 r2 q

2 4 p r2 q - r2 q2

, qF::q Ø RootB- 128 p5 r + 64 p5 r Sin@bD2 + 128 p5 - r2 I- 1 + Sin@bD2M +

K32 p4 r - 16 p4 r Sin@bD2 - 32 p4 - r2

I- 1 + Sin@bD2

M O Ò1 + 4 p2 r Ò13 - 4 p r Ò14 + r Ò15 &, 1F>,

:q Ø RootB- 128 p5 r + 64 p5 r Sin@bD2 + 128 p5 - r2 I- 1 + Sin@bD2M +

K32 p4 r - 16 p4 r Sin@bD2 - 32 p4 - r2 I- 1 + Sin@bD2M O Ò1 + 4 p2 r Ò13

- 4 p r Ò14

+ r Ò15

&, 2F>,

:q Ø RootB- 128 p5 r + 64 p5 r Sin@bD2 + 128 p5 - r2 I- 1 + Sin@bD2M +

K32 p4 r - 16 p4 r Sin@bD2 - 32 p4 - r2 I- 1 + Sin@bD2M O Ò1 + 4 p2 r Ò13

- 4 p r Ò14

+ r Ò15

&, 3F>,

:q Ø RootB- 128 p5r + 64 p5

r Sin@bD2 + 128 p5 - r2 I- 1 + Sin@bD2M +

K32 p4

r - 16 p4

r Sin@bD2 - 32 p4

- r2 I- 1 + Sin@bD2M O Ò1 + 4 p

2r Ò1

3- 4 p r Ò1

4+ r Ò1

5&, 4F>,

:q Ø RootB- 128 p5 r + 64 p5 r Sin@bD2 + 128 p5 - r2

I- 1 + Sin@bD2

M +

K32 p4

r - 16 p4

r Sin@bD2 - 32 p4

- r2 I- 1 + Sin@bD2M O Ò1 + 4 p

2r Ò1

3- 4 p r Ò1

4+ r Ò1

5&, 5F>>

RevolutionPlot3DBRootB- 128 p 5 r + 64 p 5 r Sin@bD2 +

128 p 5 - r2 I- 1 + Sin@bD2M + K32 p 4 r - 16 p 4 r Sin@bD2 - 32 p 4 - r2 I- 1 + Sin@bD2M O Ò1 +

4 p 2 r Ò13- 4 p r Ò14

+ r Ò15 &, 5F, 8r, - 10, 10<, 8b, - 20 p , 20 p <F





SolveB2 p 2 p r - 2 p r2 -

4 p r2q-r

2q2

2 p

2

q2

==4 p r2 - 2 r2 q

2 4 p r2 q - r2 q2

, qF

::q Ø 4 p

3+ 4 p

3 J- 17 + 3 33 N1ê3 - 2

3 K- 17 + 3 33 O1ê3

p>,

:q Ø4 p

3-

2 J1 + Â 3 N p

3 J- 17 + 3 33 N1ê3 +1

3K1 - Â 3 O K- 17 + 3 33 O1ê3 p>,

:q Ø4 p

3-

2 J1 - Â 3 N p

3 J- 17 + 3 33 N1ê3 +1

3K1 + Â 3 O K- 17 + 3 33 O1ê3 p>,

9q Ø p RootA- 256 + 192 Ò1 - 48 Ò12

+ 8 Ò13

- 4 Ò14

+ Ò15

&, 1E=,

9q Ø p Root

A- 256 + 192 Ò1 - 48 Ò1

2+ 8 Ò1

3- 4 Ò1

4+ Ò1

5&, 2

E=,

9q Ø p RootA- 256 + 192 Ò1 - 48 Ò12

+ 8 Ò13

- 4 Ò14

+ Ò15

&, 3E=,9q Ø p RootA- 256 + 192 Ò1 - 48 Ò12

+ 8 Ò13

- 4 Ò14

+ Ò15

&, 4E=,9q Ø p RootA- 256 + 192 Ò1 - 48 Ò12

+ 8 Ò13

- 4 Ò14

+ Ò15

&, 5E=>(4)

2 p r

q= Ish ë qM = rq; rq = light year ¹ velocity

(5)2 p

q= 1, q = 2 p

(6)

2 p r

q=

2 p r1

q+ r =

4 p r2 - 2 r2 q

2 4 p r

2

q - r

2

q

2

SolveB 2 p r1

q+ r ==

4 p r2 - 2 r2 q

2 4 p r2 q - r2 q2

, r1F

(7)::r1 Ø -

q - 2 p r2 + r2 q + r r2 H4 p - qL q

2 p r2 H4 p - qL q

>>

::r1 Ø2 p r - r q

2 p >>





SolveB 2 p r - r q

2 p ==

q - 2 p r2 + r2 q + r r2 H4 p - qL q

2 p r2 H4 p - qL q

, qF

::q Ø

28 p

15+

92 p

15 J269 + 15 Â 111 N1ê3

+2

15

K269 + 15 Â 111

O1ê3

p

>,

:q Ø28 p

15-

46 J1 + Â 3 N p

15 J269 + 15 Â 111 N1ê3 -1

15K1 - Â 3 O K269 + 15 Â 111 O1ê3 p>,

:q Ø28 p

15-

46 J1 - Â 3 N p

15 J269 + 15 Â 111 N1ê3 -1

15K1 + Â 3 O K269 + 15 Â 111 O1ê3 p>>

Plot3DB 2 p r

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

Now we can visually see how velocities would be superpositional. They can fit right on top of each other in this lego -

like system while continuing along the same contour.





RevolutionPlot3DB 2 p r

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

(8)Velocity when r is constant =4 p r2 - 2 r2 q

2 4 p r2 q - r2 q2

Plot3DB 4 p r2 - 2 r2 q

2 4 p r2 q - r2 q2

, 8r, - 1, 1<, 8q, - 4 p , 4 p <F





RevolutionPlot3DB 4 p r2 - 2 r2 q

2 4 p r2 q - r2 q2

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





RevolutionPlot3DB 4 p r2 - 2 r2 q

2 4 p r2 q - r2 q2

, 8r, - 1, 1<, 8q, - 4 p , 4 p <F

(9)

We also have access to the equation,4 p r2 - 2 r2 q

2 4 p r2 q - r2 q2

ã r ê Hq ê H2 p LL,

which winds up having the r variable cancel out,

and delivers three exact solutions for theta.

SolveB 4 p r2 - 2 r2 q

2 4 p r2 q - r2 q2

ã2 p r

q, qF





::q Ø2

32 p -

2 p

J17 + 3 33 N1ê3+ K17 + 3 33 O

1ê3

p >,

:q Ø

4 p

3 +

2 J1 + Â 3 N p

3 J17 + 3 33 N1ê3 -

1

3 K1 - Â 3 O K17 + 3 33 O1ê3

p>,

:q Ø4 p

3+

2 J1 - Â 3 N p

3 J17 + 3 33 N1ê3-

1

3K1 + Â 3 O K17 + 3 33 O

1ê3

p>>

RevolutionPlot3DBr2

32 p -

2 p

J17 + 3 33 N1ê3 + J17 + 3 33 N1ê3 p , 8r, - 1, 1<F









RevolutionPlot3DBr4 p

3+

2 J1 + Â 3 N p

3 J17 + 3 33 N1ê3 -1

3J1 - Â 3 N J17 + 3 33 N1ê3 p , 8r, - 1, 1<F





RevolutionPlot3DBr4 p

3+

2 J1 - Â 3 N p

3 J17 + 3 33 N1ê3 -1

3J1 + Â 3 N J17 + 3 33 N1ê3 p , 8r, - 1, 1<F

r Ø2 p H4 p - qL q

H4 p - qL q

4 p r2 - 2 r2 q

2 4 p r2 q - r2 q2

==2 p r

q=

2 p H4 p-qL q

H4 p-qL q

q

SolveB 4 p r2 - 2 r2 q

2 4 p r2 q - r2 q2

==

2 p H4 p-qL q

H4 p-qL q

q, rF

::r Ø -2 p

4 p2 q2 - 4 p q3 + q4

>, :r Ø2 p

4 p2 q2 - 4 p q3 + q4

>>





RevolutionPlot3DB 2 p

4 p 2 q2 - 4 p q3 + q4

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





SphericalPlot3DB 2 p

4 p 2 2 p + p 2 - p 2 Sin@bD2 2

- 4 p q3 + q4

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





SphericalPlot3DB 2 p

4 p 2 HqL2 - 4 p 2 p + p 2 - p 2 Sin@bD2 3

+ q4

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

-1.0

-0.5

0.0

0.5

1.0

-1.0

-0.5

0.0

0.5

.

-1.0

-0.5

0.0

0.5

.0





SphericalPlot3DBH2 p L ì . 4 p 2 HqL2 - 4 p HqL3 + 2 p + p 2 - p 2 Sin@bD2 4

,

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





SphericalPlot3DBH2 p L ì . 4 p

2 2 p + p 2

- p 2 Sin@bD2 2

- 4 p 2 p + p 2

- p 2 Sin@bD2 3

+ HqL4 ,

8q, - 2 p , 2 p

<,

8b, - p

ê2, p

ê2

<F

-1.0

-0.5

0.0

0.5

1.0

-1.0

-0.5

0.0

0.5

.

-1.0

-0.5

0.0

0.5

.0






2 2 p + p 2

- p 2 Sin@bD2 2

- 4 p HqL3 + 2 p + p 2

- p 2 Sin@bD2 4

,

8q, - 2 p , 2 p

<,

8b, - p

ê2, p

ê2

<F






2 HqL2 - 4 p 2 p + p 2

- p 2 Sin@bD2 3

+ 2 p + p 2

- p 2 Sin@bD2 4

,

8q, - 2 p , 2 p

<,

8b, - p

ê2, p

ê2

<F

q = 2 p + p2 - p2 Sin@bD2 =

2 p r2 + r4 - r2 h2

r2





ContourPlot3DBH2 p L ì 4 p 22 p r2 + r4 - r2 h2

r2

2

- 4 p HqL3 + HqL4 ,

8r, - 1, 1

<,

8h, - 1, 1

<,

8q, - 2 p , 2 p

<F





ContourPlot3DBH2 p L ì 4 p 2 HqL2 - 4 p

2 p r2 + r4 - r2 h2

r2

3

+ HqL4 ,

8r, - 1, 1

<,

8h, - 1, 1

<,

8q, - 2 p , 2 p

<F

-1.0

-0.5

0.0

0.5

1.0

1.0

-0.5

0.0

0.5

.

-5

0

5





ContourPlot3DBH2 p L ì 4 p 2 HqL2 - 4 p HqL3 +

2 p r2 + r4 - r2 h2

r2

4

,

8r, - 1, 1

<,

8h, - 1, 1

<,

8q, - 2 p , 2 p

<F





ContourPlot3DBH2 p L ì 4 p 22 p r2 + r4 - r2 h2

r2

2

- 4 p

2 p r2 + r4 - r2 h2

r2

3

+ HqL4 ,

8r, - 1, 1

<,

8h, - 1, 1

<,

8q, - 2 p , 2 p

<F

-1.0

-0.5

0.0

0.5

1.0

1.0

-0.5

0.0

0.5

.

-5

0

5







Plot3DBH2 p L ì 4 p 2

2 p r2 + r4 - r2 h2

r2

2

- 4 p

2 p r2 + r4 - r2 h2

r2

3

+

2 p r2 + r4 - r2 h2

r2

4

,

8r, - 1, 1<, 8h, - 1, 1<F





ContourPlot3DBH2 p L ì 4 p

2 2 p + p 2

- p 2 Sin@bD2 2

- 4 p

2 p r2 + r4 - r2 h2

r2

3

+

2 p r2 + r4 - r2 h2

r2

4

,

8r, - 1, 1<, 8h, - 1, 1<, 8b, - p , p <F






2

2 p r2 + r4 - r2 h2

r2

2

- 4 p 2 p + p 2

- p 2 Sin@bD2 3

+

2 p r2 + r4 - r2 h2

r2

4

,

8r, - 1, 1<, 8h, - 1, 1<, 8b, - p , p <F






2

2 p r2 + r4 - r2 h2

r2

2

- 4 p

2 p r2 + r4 - r2 h2

r2

3

+ 2 p + p 2

- p 2 Sin@bD2 4

,

8r, - 1, 1<, 8h, - 1, 1<, 8b, - p , p <F






2 2 p + p 2

- p 2 Sin@bD2 2

- 4 p 2 p + p 2

- p 2 Sin@bD2 3

+

2 p r2 + r4 - r2 h2

r2

4

,

8r, - 1, 1<, 8h, - 1, 1<, 8b, - p , p <F






2 2 p + p 2

- p 2 Sin@bD2 2

- 4 p

2 p r2 + r4 - r2 h2

r2

3

+ 2 p + p 2

- p 2 Sin@bD2 4

,

8r, - 1, 1<, 8h, - 1, 1<, 8b, - p , p <F






2

2 p r2 + r4 - r2 h2

r2

2

- 4 p 2 p + p 2

- p 2 Sin@bD2 3

+ 2 p + p 2

- p 2 Sin@bD2 4

,

8r, - 1, 1<, 8h, - 1, 1<, 8b, - p , p <F

ü A Few thoughts on the Wavelength with Changing Value

h =4 p r2 q - r2 q2

2 p

r Ø2 p H4 p - qL q

H4 p - qL q

DB4 p

2 p H4 p-qL q

H4 p-qL q

2

q -2 p H4 p-qL q

H4 p-qL q

2

q2

4 p 2, qF

16 p3

H4 p-qL2-

4 p2

4 p-q-

4 p2q

H4 p-qL2

8 p2 16 p3

4 p-q-

4 p2q

4 p-q





RevolutionPlot3DB 16 p 3

H4 p-qL2-

4 p 2

4 p-q-

4 p 2q

H4 p-qL2

8 p 216 p

3

4 p-q-

4 p 2q

4 p-q

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

H1 ê H2 p LL DB 16 p 3

H4 p-qL2-

4 p 2

4 p-q-

4 p 2q

H4 p-qL2

8 p 216 p

3

4 p-q-

4 p 2q

4 p-q

, qF

-

16 p3

H4 p-qL2-

4 p2

4 p-q-

4 p2q

H4 p-qL2

2

16 p2

16 p3

4 p-q-4 p2 q

4 p-q

3ê2 +

32 p3

H4 p-qL3-

8 p2

H4 p-qL2-

8 p2q

H4 p-qL3

8 p2

16 p3

4 p-q-4 p

2q

4 p-q

2 p





RevolutionPlot3DB-

16 p 3

H4 p-qL2-

4 p 2

4 p-q

-4 p

2q

H4 p-qL2

2

16 p 2

16 p 3

4 p-q

-4 p

2q

4 p-q

3ê2+

32 p 3

H4 p-qL3-

8 p 2

H4 p-qL2-

8 p 2q

H4 p-qL3

8 p 2

16 p 3

4 p-q

-4 p

2q

4 p-q

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


Date post:	09-Apr-2018
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10 Dimensions: The Wavelength with Constant Value by Parker Emmerson ©2009-2010

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