Why the 2-generator subgroup is interesting:
because it resembles closely a two-body system. In $$ \mathit R $$ the set of relations for this subgroup, which is any two successive terms in the cyclic permutation of A = URFDLB.
There are two 4-dimensional numbers or elements of $$ M_4 $$, and the relations for an individual 'free' element $$ g \in G$$, i.e. any singleton element $$ X \in A $$, has a signature (mod 4).
Two co-rotating "frames" $$ Xg_1, \,Yg_2 $$ will generate a pressure which corresponds to a $$ dW/dt $$.
Consider phases of the lunar orbit, which are independent of the position of the earth in space, but depend only on the gravitational energy of the local earth-moon system.
Assign the beginning and ending phases called "new" and "full" to states $$s_i $$, that represent even divisors of some lunar function (of being "full"), and the phases "first quarter" and "last quarter" to odd-i states. So you quarter the phases of the moon as $$ s_1, s_2, s_3, s_4 $$ and assign "the identity" to say, i=4.
Now define a "j-space" which maps each state to a coordinate system (say, on the surface of earth); this assumes a "first derivative" for each state exists in time and space.as much as there 'will be' a first appearance (in the map) of a lighted crescent, as the moon moves out of the earth's shadow - this event, for example, measures the occurrence of the end of the "new" phase).
So there are two 4-dimensional numbers, interacting in space and time and there's a record of these phases (quite a long record, really), in the space and time (and even dummies can see that 3 + 1 add up to 4).
because it resembles closely a two-body system. In $$ \mathit R $$ the set of relations for this subgroup, which is any two successive terms in the cyclic permutation of A = URFDLB.
There are two 4-dimensional numbers or elements of $$ M_4 $$, and the relations for an individual 'free' element $$ g \in G$$, i.e. any singleton element $$ X \in A $$, has a signature (mod 4).
Two co-rotating "frames" $$ Xg_1, \,Yg_2 $$ will generate a pressure which corresponds to a $$ dW/dt $$.
Consider phases of the lunar orbit, which are independent of the position of the earth in space, but depend only on the gravitational energy of the local earth-moon system.
Assign the beginning and ending phases called "new" and "full" to states $$s_i $$, that represent even divisors of some lunar function (of being "full"), and the phases "first quarter" and "last quarter" to odd-i states. So you quarter the phases of the moon as $$ s_1, s_2, s_3, s_4 $$ and assign "the identity" to say, i=4.
Now define a "j-space" which maps each state to a coordinate system (say, on the surface of earth); this assumes a "first derivative" for each state exists in time and space.as much as there 'will be' a first appearance (in the map) of a lighted crescent, as the moon moves out of the earth's shadow - this event, for example, measures the occurrence of the end of the "new" phase).
So there are two 4-dimensional numbers, interacting in space and time and there's a record of these phases (quite a long record, really), in the space and time (and even dummies can see that 3 + 1 add up to 4).