Seven Circles: The Fano Plane Relaxed to 7 Dimensions

OCTONIONS

I preferred not to mar the look by arrows but the cyclic order is implied by noting e.g. that the redGreenAqua crossing marks the far end of basic vector e1. Similarly for e2 e3 e4 e5 e6 e7. These being orthonormal in the object depicted, you are looking at something in real Euclidean 7-dimensional space! The stereoptic fusion lifts the flat, 2d ellipses to spatial circles, if you allow the usual few seconds for accomodation. Each circle meets the other six, two at each of three equally spaced points---e.g. the red circle meets green and aqua at point e1, the blue and brown at e2, the pink and yellow at e3. The easiest figure I call ``the steering wheel'', Steering Wheel got by looking along the 1110000 direction, so that the red circle is a great round circle, the other six circles are however edge-on line segments, half hidden behind another, so you actually see only four of the seven colors there. The lines though develop into narrow ellipses by adding a small random 7vector to 1110000, in another view, 'Steering Wheel' with small Random Factor---though one of the ellipses here is for me still indistinguishable from a stick. This is a view, Random View in which the viewing direction is a random selection. How is this a multiplication table? Circle 123 signifies that e1 e2 e3 behave like the i j k of quaternions, namely, that ij=k jk=i ki=j. Each e squares to -1. All seven e's anticommute, e.g. ji=-k . 123 is code for e1 e2 = e3, e2 e3 = e1, e3 e1 = e2. Beyond this the figure tells us so many e_number times e_othernumber products that we can multiply any pair of them. (e0 would be the real number 1 so squares to +1; octonions have an ordinary real part, like complex numbers, but the space of imaginary parts is real 7-dimensional, which is why we show seven e_subscript vectors rather than eight.) Perhaps we should also show the 0000000 zero-imaginary-part point as an extra black thing?---so far I've hesitated for fear of marring the pretty circles with a roach. Texts--physics:SusumuOkubo.CambridgeUPress--math:JohnHConway&DerekASmith,AKPe\ ters,WellesleyMASS--octonion(s) in both titles