4 (CB)  4 (CB)  6  7 (AB)  6  8 (BH)  7 (GS)  4 (AS)  6 (AS)  
4  8  5 (AS)  4 (GS)  10 (CB)  4 (AS)  5 (GS)  6 (AB)  
4  6 (GS)  7 (GS)  10 (GS)  6 (GS)  4 (GS)  7 (GS)  
9 (JD)  8 (JD)  13 (GS)  10 (GS)  7 (GS)  6 (AS)  
7 (JD)  13 (GS)  10 (GS)  5 (JD)  8 (AB)  
10 (GS)  7 (GS)  6 (GS)  8 (GS)  
10 (CB)  10 (GS)  14 (GS)  
8 (GS)  9 (GS)  
4 (AS) 
Here are all the other best known nsymbiotic tilings for n ≤ 6:

Andrew Bayly found the following symbiotic tilings:









Joe DeVincentis found this 2symbiotic tiling:

Bryce Herdt found this symbiotic tiling:

Bryce Herdt also found an infinite family of symbiotic tilings of trapezoids and triangles that works for all even n≥6:

Andrew Bayly conjectured that an nsymbiotic pair exists for every n>1, though he was only able to prove this for even n. Then Joe DeVincentis provided the argument for odd n.
Joe DeVincentis also noted that the tan and tritan featured above with symbiosis number 3 is also nsymbiotic for all n≥5. Bryce Herdt noted that this pair is also 4symbiotic.
Bryce Herdt also proved that any 2symbiotic pair is also nsymbiotic for all n≥3.
Jeremy Galvagni wonders whether every positive integer n>1 is the symbiosis number for some pair of shapes. The smallest n which is in doubt is n=10.
Joshua Taylor also played around with symbiosis numbers of "sliced polyominoes":

If you can extend any of these results, please email me. Click here to go back to Math Magic. Last updated 2/10/13.