The Circle That Returns
Meditations on Oneness · Local folio mark 1111111
Luna's sixfold window
At the Observatory, Luna records a brass window whose six markers are spaced by equal turns. For an angle , write
This is the point at angle on the unit circle. Angles differing by name the same point.
Field plate · Rules for this investigation
Multiplication adds turns
Let . Then and .
Find every sixth root
Label the window here. Use separate paper for the proof.
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Label the six vertices with and . Prove that these six points are all the solutions of .
For the “no others” part, write . What must force and to be?
Luna's trace record · Repeated multiplication
Which roots visit the whole circle?
Because , exponents differing by 6 name the same root. For example, .
Starting at , repeatedly multiply by the root in the first column. Record the exponents visited before returning to 0.
Some traces close early. The complete state record will show which ones.
Trace the cycles
Complete the cycle record on this page.
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A root is a generator when its repeated powers visit all six roots.
Complete the table. Which roots are generators?
Multiplier Exponents visited, beginning with 0 First return after Visits all six?
Archive synthesis · One object, one operation
When a set becomes a group
A group is a set with an operation having these properties:
- multiplying two members stays in the set;
- parentheses do not change the result;
- one identity leaves every member unchanged;
- every member has an inverse that multiplies with it to give the identity.
You may use the fact that complex multiplication is associative.
Prove the hidden structure
Use separate paper for your group proof.
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Prove that the six solutions of form a group under multiplication.
Use and reduce exponents modulo 6. Name the identity and the inverse of every . Finally, explain why this group is cyclic: one root’s repeated powers produce every root.