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Kea

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Regarding

http://math.ucr.edu/home/baez/week214.html

...I tried to post something to the SPR thread but I guess there were too many equations. Anyway, here's a PF thread. After reading TWF, I thought of the following quote:

This quote appears in the book

These are the sorts of lattices that quantum logicians like.

Tony Smith, on the page linked by John Baez, mentions the Golden Ratio in connection with the Fano plane. John has mentioned Fibonacci numbers. I thought it would therefore be interesting to bring up the following observations.

In the paper

J.W. Barrett, http://xxx.lanl.gov/abs/gr-qc/0203018

John Barrett discusses the following remarkable Fourier transform for [itex]6j[/itex] symbols

[tex]\frac{1}{N} \sum_{j_{1} \cdots j_{6}} [ j_{4}, j_{5}, j_{6};

j_{1}, j_{2}, j_{3} ]^{2} H(j_{1},i_{1}) \cdots H(j_{6},i_{6}) = [

i_{1}, i_{2}, i_{3}; i_{4}, i_{5}, i_{6} ]^{2}[/tex]

[itex]N[/itex] is a normalisation constant. Choosing [itex]q =

e^{\frac{i \pi}{5}}[/itex] as in

M. Freedman M. Larsen Z. Wang,

http://arxiv.org/abs/quant-ph/0001108

gives allowable spin values [itex]j \in 0, \frac{1}{2}, 1, \frac{3}{2}[/itex]. The kernel function is given by the Hopf link invariant

[tex]H(j,i) = (-1)^{2i + 2j} \cdot \frac{\sin \frac{\pi}{5}(2j +

1)(2i + 1)}{\sin \frac{\pi}{5}}[/tex]

Observe that [itex]H[/itex] only takes values [itex]\pm \phi , \pm 1[/itex] where [itex]\phi = 1.61803399 \cdots[/itex] is the golden ratio

[tex]\phi = \frac{\sin \frac{2 \pi}{5}}{\sin \frac{\pi}{5}}[/tex]

The full kernel therefore takes values in powers of [itex]\phi[/itex], or in other words the truncated Fibonacci sequence

[tex]1 , \phi , \phi + 1 , 2 \phi

+ 1 , 3 \phi + 2 , 5 \phi + 3 , 8 \phi + 5[/tex]

of 7 terms.

Cheers

Kea

**This Week's Finds 214**http://math.ucr.edu/home/baez/week214.html

...I tried to post something to the SPR thread but I guess there were too many equations. Anyway, here's a PF thread. After reading TWF, I thought of the following quote:

*Continuous geometries...are a generalization of complex projective geometry somewhat in the way that Hilbert space is a generalization of finite dimensional Euclidean space***Halperin**(1960)This quote appears in the book

*Orthomodular Lattices*by G. Kalmbach, on page 191. Just below this quote is the example of the Fano plane and its associated complete modular ortholattice on 16 points.These are the sorts of lattices that quantum logicians like.

Tony Smith, on the page linked by John Baez, mentions the Golden Ratio in connection with the Fano plane. John has mentioned Fibonacci numbers. I thought it would therefore be interesting to bring up the following observations.

In the paper

*Geometrical measurements in three dimensional quantum gravity*,J.W. Barrett, http://xxx.lanl.gov/abs/gr-qc/0203018

John Barrett discusses the following remarkable Fourier transform for [itex]6j[/itex] symbols

[tex]\frac{1}{N} \sum_{j_{1} \cdots j_{6}} [ j_{4}, j_{5}, j_{6};

j_{1}, j_{2}, j_{3} ]^{2} H(j_{1},i_{1}) \cdots H(j_{6},i_{6}) = [

i_{1}, i_{2}, i_{3}; i_{4}, i_{5}, i_{6} ]^{2}[/tex]

[itex]N[/itex] is a normalisation constant. Choosing [itex]q =

e^{\frac{i \pi}{5}}[/itex] as in

*A modular functor which is universal for quantum computation*,M. Freedman M. Larsen Z. Wang,

http://arxiv.org/abs/quant-ph/0001108

gives allowable spin values [itex]j \in 0, \frac{1}{2}, 1, \frac{3}{2}[/itex]. The kernel function is given by the Hopf link invariant

[tex]H(j,i) = (-1)^{2i + 2j} \cdot \frac{\sin \frac{\pi}{5}(2j +

1)(2i + 1)}{\sin \frac{\pi}{5}}[/tex]

Observe that [itex]H[/itex] only takes values [itex]\pm \phi , \pm 1[/itex] where [itex]\phi = 1.61803399 \cdots[/itex] is the golden ratio

[tex]\phi = \frac{\sin \frac{2 \pi}{5}}{\sin \frac{\pi}{5}}[/tex]

The full kernel therefore takes values in powers of [itex]\phi[/itex], or in other words the truncated Fibonacci sequence

[tex]1 , \phi , \phi + 1 , 2 \phi

+ 1 , 3 \phi + 2 , 5 \phi + 3 , 8 \phi + 5[/tex]

of 7 terms.

Cheers

Kea

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