Math 2.0

Geometry Definition

Bachmann's reflection geometry

Builds on GroupReflectionAbsolute geometry

Definition

Let GG be a group and let SS be a set of involutions of GG, that is, of elements of order 22. Assume that SS generates GG and is closed under conjugation by elements of GG. Call the elements of SS lines, written a,b,c,g,h,ja, b, c, g, h, j. For σ,τ∈G\sigma, \tau \in G, write σ∣τ\sigma \mid \tau if στ\sigma \tau is an involution. Call the elements abab with a,b∈Sa, b \in S and a∣ba \mid b points, written P,QP, Q. For a point PP and a line gg, P∣gP \mid g means that PP lies on gg. For lines aa and bb, a∣ba \mid b means that they are perpendicular. The pair (G,S)(G, S) is a metric plane if:

  1. for any points P,QP, Q there is a line gg with P∣gP \mid g and Q∣gQ \mid g;
  2. if P∣gP \mid g, P∣hP \mid h, Q∣gQ \mid g and Q∣hQ \mid h, then P=QP = Q or g=hg = h;
  3. if a,b,c∣Pa, b, c \mid P for a point PP, then abc∈Sabc \in S;
  4. if a,b,c∣ga, b, c \mid g for a line gg, then abc∈Sabc \in S;
  5. there are lines g,h,jg, h, j with g∣hg \mid h such that none of j∣gj \mid g, j∣hj \mid h, j∣ghj \mid gh holds.

Remarks

Clauses 3 and 4 are the three reflections theorems. The product of the reflections in three lines through a common point, or in three lines perpendicular to a common line, is again a reflection. Clause 5 excludes degenerate cases.

Example: let GG be the group of isometries of the Euclidean plane generated by the reflections in lines, and SS the set of these reflections. Two reflections compose to an involution exactly when their lines are perpendicular, and then the product is the half-turn about the intersection point. A half-turn about PP times the reflection in gg is an involution exactly when PP lies on gg. The product of reflections in three concurrent lines, or in three parallel lines, is a reflection, which gives clauses 3 and 4.

Non-example: let GG be cyclic of order 22 and SS its non-identity element ss. Then ss=1s s = 1 is not an involution, so there are no points and clause 5 fails.

Metric planes give a group-theoretic form of plane absolute geometry. The real hyperbolic plane and the real elliptic plane give further examples, with the same construction from their reflections.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

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Further reading

  1. Web https://en.wikipedia.org/wiki/Absolute_geometry
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DefinesBachmann's reflection geometry
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