Re: Hexagonal Spiral

• To: mathgroup at smc.vnet.net
• Subject: [mg56962] Re: Hexagonal Spiral
• From: Paul Abbott <paul at physics.uwa.edu.au>
• Date: Wed, 11 May 2005 05:24:29 -0400 (EDT)
• Organization: The University of Western Australia
• References: <d5mub1\$dpm\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```In article <d5mub1\$dpm\$1 at smc.vnet.net>, zak <chocolatez at gmail.com>
wrote:

> the purpose of this message is to draw a hexagonal spiral and then
> divide every edge to the suitable numbers of points, then map the
> prime numbers to the all points of the spiral.

In a recent posting the following code for creating a spiral on an
integer lattice was presented.

IntegerSpiral[n_] := {Re[#],Im[#]}& /@
Fold[Join[#1, Last[#1]+I^#2 Range[#2/2]]&, {0}, Range[n]]

The idea is that the length of the limbs of the spiral follow the
pattern 1,1,2,2,3,3,4,4,5,5,... and that each limb is at a 90 degree
angle to the previous limb (multiplication by I in the complex plane).

Changing I to Exp[I Pi/3] means that each limb is at a 60 degree angle
to the previous limb -- a hexagonal spiral:

HexagonalSpiral[n_] := {Re[#],Im[#]}& /@
Fold[Join[#1, Last[#1]+Exp[I Pi/3]^#2 Range[#2/2]]&, {0}, Range[n]]

One can use

spiral = HexagonalSpiral[100]

to construct the spiral,

primes = spiral[[ Prime[Range[PrimePi[Length[spiral]]]] ]];

to find the prime points, and

ListPlot[spiral, PlotJoined->True, Axes -> False,
AspectRatio->Automatic,
Epilog -> {PointSize[0.02], Point /@ primes}];

to draw the spiral and display those with prime index as dots.

> this is motivated by the figure:
> http://www.cut-the-knot.org/ctk/HexMosaic.gif
> in an article in:
> http://www.cut-the-knot.org/ctk/FromLCarrollToArchimedes.shtml

Tweaking might be needed so as to get the above code to exactly
correspond to this figure.

Cheers,
Paul

--
Paul Abbott                                      Phone: +61 8 6488 2734
School of Physics, M013                            Fax: +61 8 6488 1014
The University of Western Australia         (CRICOS Provider No 00126G)
AUSTRALIA                               http://physics.uwa.edu.au/~paul
http://InternationalMathematicaSymposium.org/IMS2005/

```

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