by **Marek14** » Fri Jun 20, 2014 8:52 am

Now that I think about it, maybe the spheritorus/ditorus lock could be undone by rotating the spheritorus instead of translating? This might put some additional constraints on numbers, but the system is still good.

So, how about ditorus/ditorus combinations?

Here, we'll have a 3x3 matrix of 9 variables. We'll start from ((1,0,0),(0,0,0),(0,0,0)) and from a chain of four A's and four B's.

[pause for writing a program to write the list/eliminate equivalent entries]

Also, maybe "lock" can be more easily defined as stemming from a chain with same letter on both ends?

I derived 8 basic groups of double-ditorus positionings, 5 of which should be doable in 4D.

Group 1:

AAABABBB - major/minor, medium/minor, minor/major, minor/medium. ((1,0,a),(0,0,b),(c,d,0)). (a+1),b,(c+d)-ditorus and (c+1),d,(a+b)-ditorus. Two ditoruses linked through their outer holes. Note that this configuration isn't possible until 5D where you can have two 212-ditoruses link this way.

AAABBBAB - alternate configuration

ABBBAAAB - another alternate configuration. This is probably impossible since the ditoruses wouldn't fit this way -- each would have to be bigger than the other. Could be done with asymmetrical ditoruses, though.

AABABABB - This time, they are threaded through both outer and inner holes. Looks topologically distinct.

AABBABAB - alternate configuration

ABAABBAB - another alternate configuration

ABABBAAB - yet another?

AABABBBA - Here, one ditorus goes through outer hole of the other, which goes through both of the first torus's holes. A lock.

AABBBABA - alternate configuration

ABABABAB - Ultimate threading!

ABABABBA - Another lock.

Group 2:

AAABBABB - major/medium, major/minor, medium/medium, medium/minor, minor/major. ((1,a,b),(0,c,d),(e,0,0)). (a+b+1),(c+d),e-ditorus and (e+1),(a+c),(b+d)-ditorus. Two ditoruses linked through outer hole of one and inner hole of other. If (a,d) or (b,c) is 0, this can be done with two ditoruses in 4D.

AABBBAAB - alternate configuration

AABBABBA - This is a lock where second ditorus goes through both holes of the first.

Group 3:

AABABBAB - major/minor, medium/major, medium/minor, minor/major, minor/medium. ((1,0,a),(b,0,c),(d,e,0)). (a+1),(b+c),(d+e)-ditorus and (b+d+1),e,(a+c)-ditorus. Here, the outer hole of one ditorus is locked in outer hole of the other. If c and d is 0, this can be achieved in 4D.

Group 4:

AABBAABB - major/medium, major/minor, medium/major, minor/major. ((1,a,b),(c,0,0),(d,0,0)). (a+b+1),c,d-ditorus and (c+d+1),a,b-ditorus. The ditoruses are linked through their inner holes. Can't be done until two 311-ditoruses in 5D.

Group 5:

ABAABBBA - major/major, major/medium, major/minor, medium/minor, minor/major, minor/medium. ((a+1,b,c),(0,0,d),(e,f,0)). (a+b+c+1),d,(e+f)-ditorus and (a+e+1),(b+f),(c+d)-ditorus. Outer hole of one is locked through both holes of the other. If (a,c,f) or (b,c,e) is 0, this can be done in 4D.

Group 6:

ABABBABA - major/major, major/minor, medium/minor, minor/major, minor/medium. ((a+1,0,b),(0,0,c),(d,e,0)). (a+b+1),c,(d+e)-ditorus and (a+d+1),e,(b+c)-ditorus. With a = 0, this lock is possible in 5D for two 212-ditoruses and with b = 0 or d = 0 for 212-ditorus and 311-ditorus.

Group 7:

ABBAABBA - major/major, major/medium, major/minor, medium/medium, medium/minor, minor/major. ((a+1,b,c),(0,d,e),(f,0,0)). (a+b+c+1),(d+e),f-ditorus and (a+f+1),(b+d),c-ditorus. If (a,b,e) is 0, this is possible in 4D.

Group 8:

ABBABAAB - major/minor, medium/medium, medium/minor, minor/major, minor/medium. ((1,0,a),(0,b,c),(d,e,0)). (a+1),(b+c),(d+e)-ditorus and (d+1),(b+e),(a+c)-ditorus. If (c,e) is 0, this is possible in 4D.

Degenerates:

AAAABBBB - no chain.

AAABBBBA - A has no functional minor dimensions.

AABBBBAA - A has no functional minor dimensions.

Group 2 model:

AAABBABB, (a,d)=0:

((x,0,y),(0,z,0),(w,0,0))

(sqrt((sqrt(x^2 + y^2) - 8)^2 + z^2) - 6)^2 + w^2 = 1

(sqrt((sqrt((x-14)^2 + w^2) - 6)^2 + z^2) - 2)^2 + y^2 = 1

x [-15,23]; y [-15,15]; z [-7,7]; w [-9,9]

AAABBABB, (b,c)=0:

((x,y,0),(0,0,z),(w,0,0))

(sqrt((sqrt(x^2 + y^2) - 8)^2 + z^2) - 6)^2 + w^2 = 1

(sqrt((sqrt((x-14)^2 + w^2) - 6)^2 + y^2) - 2)^2 + z^2 = 1

x [-15,23]; y [-15,15]; z [-7,7]; w [-9,9]

AABBBAAB (or rather BBAAABBA), (a,d)=0:

((x,0,y),(0,z,0),(w,0,0))

(sqrt((sqrt(x^2 + y^2) - 8)^2 + z^2) - 6)^2 + w^2 = 1

(sqrt((sqrt((x+6)^2 + w^2) - 14)^2 + z^2) - 2)^2 + y^2 = 1

x [-23,15]; y [-15,15]; z [-7,7]; w [-17,17]

AABBBAAB (or rather BBAAABBA), (b,c)=0:

((x,y,0),(0,0,z),(w,0,0))

(sqrt((sqrt(x^2 + y^2) - 8)^2 + z^2) - 6)^2 + w^2 = 1

(sqrt((sqrt((x+6)^2 + w^2) - 14)^2 + y^2) - 2)^2 + z^2 = 1

x [-23,15]; y [-15,15]; z [-7,7]; w [-17,17]

AABBABBA, (a,d)=0:

((x,y,0),(0,0,z),(w,0,0))

(sqrt((sqrt(x^2 + y^2) - 12)^2 + z^2) - 6)^2 + w^2 = 1

(sqrt((sqrt((x-6)^2 + w^2) - 6)^2 + y^2) - 2)^2 + z^2 = 1

x [-19,19]; y [-19,19]; z [-7,7]; w [-9,9]

AABBABBA, (b,c)=0:

((x,0,y),(0,z,0),(w,0,0))

(sqrt((sqrt(x^2 + y^2) - 12)^2 + z^2) - 6)^2 + w^2 = 1

(sqrt((sqrt((x-6)^2 + w^2) - 6)^2 + z^2) - 2)^2 + y^2 = 1

x [-19,19]; y [-19,19]; z [-7,7]; w [-9,9]

Group 3 model:

AABABBAB, (c,d)=0:

((x,0,y),(z,0,0),(0,w,0)

(sqrt((sqrt(x^2 + y^2) - 10)^2 + z^2) - 6)^2 + w^2 = 1

(sqrt((sqrt((x-10)^2 + z^2) - 6)^2 + w^2) - 4)^2 + y^2 = 1

x [-17,21]; y [-17,17]; z [-11,11]; w [-5,5]

Group 5 model:

ABAABBBA, (a,c,f)=0:

((x,y,0),(0,0,z),(w,0,0))

(sqrt((sqrt(x^2 + y^2) - 14)^2 + z^2) - 12)^2 + w^2 = 1

(sqrt((sqrt((x-10)^2 + w^2) - 8)^2 + y^2) - 6)^2 + z^2 = 1

x [-27,27]; y [-27,27]; z [-13,13]; w [-15,15]

To be continued...