Biradical numbers are those numbers that are obtained recursively from 1 by addition x+y , multiplication x.y , negativation -x , reciprocation 1/x for x not 0 and biradication sr(x) of nonnegative numbers, meaning square root (contrary to the Bavarian notation of the included publications here conventional notation is used for functions).
Felix-numbers are the biradical numbers inside the interval from -1 to 1 (without the limits).
Klein-numbers are constructed on the basis of the Euclidean Klein-model of Lobachevsky geometry. Klein-numbers are biradical numbers that are obtained recursively:
starting from sr(2sr(2)-2) by successive application of
evition x.y/(y+z) where (x.x+y.y)<1 and not y+z=0
clarition sr((x.x-y.y)/(1-y.y)) where not x.x<y.y
jacition (1-sr((1-y.y.(1-x.x))/(1-z.z)))/x where not x=0 , else equal 0
Klein-numbers lie inside the interval from -1 to 1 (without the limits) by construction.
Klein-numbers are Felix-numbers.
Conjecture: not all Felix-numbers are Klein-numbers
It looks quite obvious, however, a proof is needed. It is sufficient to show that 1/2 is not a Klein-number. If you have a proof, please contact the author !
With two Felix-numbers as coordinates one gets the points of the Felix-circle
With two Klein-numbers as coordinates one gets the points of the Klein-circle.