Meeting Notes 2024-02-27

This Week’s Finds

We had quite a few interesting finds this week.

Polygon Affine Reflection Dynamics

Lael introduced an action on the set of polygons that, given a \(n\)-gon \(P\), produces an \(m\)-gon \(P’\) where \(m \leq n\). This function \(f: P \mapsto P’\) can be described algorithmically:

• Choose a vertex \(v_i\) from the original polygon \(P\)
• Fine the line segment \(\ell_i\) opposite the vertex \(v_i\) and create a new line \(\ell_i’\) through \(v_i\) such that \(\ell_i\) and \(\ell_i’\) are parallel
• Repeat for all remaining vertices
• Define new line segments by the intersections of pairs of adjacent new lines \(\ell_i’\) and \(\ell_{i+1}’\)
• Define a new polygon \(P’\) by the intersecting line segments defined above
• Rescale \(P’\) so that the area is preserved,  \(A(P) = A(P’)\)

It was suggested that this function introduces a flow on the space of polygons, with interesting behavior and potential limit points. A specific example, \(7\)-gons seemed to converge to triangles. More on this soon.

 

Measures of a Polygon’s Convexity and Regularity

This topic had a fortunate collision, both Lael and Jim had encountered the problem of constructing a measure for polygon convexity recently. Lael introduced a measure by computing a sum of cross products, where the angles are determined by intersections of opposing line segments. Jim introduced the measure known as polygonal entropy [1]. This method constructs a probability density on the polygon by asking the question “how much of the polygon is visible from each point \(p in P\)”, then computes the Shannon entropy of the distribution. Here, “visible” refers to all points \(q \in P\) such that the straight line connecting \(p\) and \(q\) stays “inside” the polygon. A brief proof was given that shows the entropy is maximized for fully-convex polygons, since every point can “see” every other point. Jim motivated the discussion of convexity by relating it to the problem of algorithmically detecting the boundary of an arbitrary subspace, which has shown up in his research recently.

 

Riemann Tensor Symmetries (Update)

Last week we discussed the symmetries of the Riemann tensor indices being related to a subset of D8 (the dihedral group of symmetries of the square), noting that not all of the reflection operations seemed to apply. This week, Jim showed that the group generated by the Riemann symmetries (denoted \(Y(2+2)\) due to relation with the Young symmetrizer) is actually isomorphic to D8. The convention of labelling vertices of the square clockwise made this more difficult to see.

 

Focus Session:

This focus session was largely a recap of the week’s reading, which included symmetric groups, group actions, and group orbits.

 

Symmetric Groups

We discussed the definition of symmetric groups [2], with the example of \(S_3\). We connected this example to last weeks focus session where we discussed the Dihedral groups, and showed that \(D_8 \cong S_3\). Further, we worked a few examples of symmetric group element multiplication using cycle notation [3] and functional composition.

 

Group Actions and \(SO(3)\)

Joshua gave an example of a group action by discussing \(SO(3)\), the Lie group of 3-dimensional rotations, and its natural action of \(S^2\), the 2-sphere. For a given radius \(r\), the group of rotations can take any point on the \(2\)-sphere of radius \(r\) to any other point. In this fashion, the group orbits of \(SO(3)\) are diffeomorphic to \(S^2\) (spherical shells).

 

Up Next:

Interesting or Relevant Talks:

• Spring Break (woot!)
• Math Department Colloquium: Extremal and Near-Extremal Black Holes: link
  • Thursday at 3:30pm – 4:30pm, 114 McAllister
• Ashtekar Frontiers of Science Lecture 6: Einstein, Gravitational waves, black holes, and other matters: link
  • Saturday at 11:00am – 12:30pm, 100 Thomas

References:

1. H. I. Stern, Polygonal Entropy: A Convexity Measure, Pattern Recognition Letters 10, 229 (1989).
2. Symmetric Groups: https://en.wikipedia.org/wiki/Symmetric_group
3. Cycle Notation: https://en.wikipedia.org/wiki/Permutation#Cycle_notation