Automorphisms of the Fano plane on the Klein quartic

Warning: program very slow to load; a bit better once loaded. Code is not very efficient. This does not work on all phones, but it works on my Samsung. For most devices you'll probably want to immediately change to landscape mode. layout: portrait/landscape (change for wide enough screens)

Note on controls:

Most of the controls are for my own use for illustrating different concepts, but it's easier to have only one program rather than two.

• "time" just controls the position of a point passing along the Hamiltonian cycle; "before" this point is coloured, "after" is black
• "widthH" is the width of the black Hamiltonian path line
• "widthHC" is the width of the colour part of Hamiltonian path line
• "widthHB" is the width of the black lines bordering hexagons
• "widthHB2" is the width of the white lines bordering hexagons
• "widthB1" and "widthB2" are for the boundary of the triangles
• Colour option: (1) is my first choice of bright colours; (2), (3), (4) are an attempt to give a choice for people who might be colour blind. (5) shows one choice black, the others pale, so the movement of the black line can be more easily picked out. (6) is a random choice; you'll get different values each time you choose this. (7) is a choice to be used with disc pattern (2), where discs are solid colours, and red, blue, green are a basis, so we can see how the other discs on a line are determined by the sum of the colours, e.g., on the line with red, blue discs, the third disc is magenta, since red+blue = magenta (light addition).
• Disc patterns (1) each disc is on three lines; the dots on the discs correspond to the colours of these three lines. (2) is a dual version, where the line colours are determined by the disc colours. (3) is a version where the discs have a larger border to more easily track their movement (note, one of the radius sliders can change the disc size). (4) Colour is average of line colours disc lies on.
• Disc numbers: Various configurations; maybe it's interesting to see the path of just one disc. Maybe it's interesting to see how x number move and interact. Maybe it's interesting to look at how colinearity or non colinearity is preserved; maybe basis or projective frame of reference is interesting. So various options.
• I will try and get round to adding more notes at some point

The Fano plane

The Fano plane is a collection of 7 points and 7 lines. Each line has three points on it, each point is on three lines, every two lines intersect at some point, every two points are on a common line.

The automorphisms

An automorphism of the Fano plane is a way of switching round the points in such a way that if points were on a common line before switching, they are still on a common line after switching, etc; all the relationships are the same after switching as before. There are 168 such automorphisms. I've put these into a game here. There are lots of websites dicussing the Fano plane and its automorphisms. E.g. Wikipedia.
An article by Shintaro Fushida-Hardy describing a crochet Klein quartic and automorphisms of the Fano plane can be found here (same article here.

The Cayley graph

A graph can be drawn of the elements of the automorphism group. This is graph in the sense of vertices (points), joined by edges: Draw a dot for each element of the automorphism group; Choose some generators, and connect the dots by directed arrows if they are related by multiplication by one of the chosen generators. For more details, see Wikipedia and group theory and finite geometry textbooks and web pages, etc, e.g., Group Properties Wik.
Choose different generators, and you'll get a different graph. There is a nice example at Webb.
Every element of the automorphism group is listed on this wikiversity page.
I've taken two generators, one of order 2, and the other of order 7. I've chosen 21 different possible pairs of generators. you can change these with the given slider.
Now use the fact that the automorphism groups of the Fano plane and the Klein quartic are isomorphic. See the Wikipedia page on the Klein quartic. This program illustrates an example of an isomorphism between the automorpshim groups. For the Klein quartic, each automorphism is uniquely determined by the image of a chosen triangle, so the triangles in the Klein quartic picture correspond to elements of the automorphism group, so give an easy way to draw a Cayley graph.
I want to step through the elements of the automorphism group in a nice way. Nice is subjective; for me, nice means working my way through a Hamiltonian cycle on a Cayley graph.