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I know (or, rather, believe) that it remains unknown whether every polygon has a periodic billiard path. But Masur proved in the 1980's that every rational polygon (vertex angles rational multiples of $\pi$) has (many) periodic billiard paths.

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      ![Hooper][1]
      ^{(Image from: W. Patrick Hooper, "Some irrational polygons have many periodic billiard paths." [PDF download link](http://wphooper.com/docs/talks/2008/milwaukee_printable.pdf).)}

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My question concerns *simple* paths: non-self-intersecting, i.e., embedded paths:

Q. Is it known that certain classes of rational polygons have simple periodic billiard paths? If so, which? That certain classes (of rational polygons) have no such simple periodic paths? If so, which?

For example, in an acute triangle, its pedal triangle is a simple periodic path, and the only such.

To be narrowly specific: Might every regular polygon have a simple periodic path?

I know (or, rather, believe) that it remains unknown whether every polygon has a periodic billiard path. But Howard Masur proved in the 1980's that every rational polygon (vertex angles rational multiples of $\pi$) has (many) periodic billiard paths.

---

      ![Hooper][1]
      ^{(Image from: W. Patrick Hooper, "Some irrational polygons have many periodic billiard paths." [PDF download link](http://wphooper.com/docs/talks/2008/milwaukee_printable.pdf).)}

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My question concerns *simple* paths: non-self-intersecting, i.e., embedded paths:

Q. Is it known that certain classes of rational polygons have simple periodic billiard paths? If so, which? That certain classes (of rational polygons) have no such simple periodic paths? If so, which?

For example, in an acute triangle, its pedal triangle is a simple periodic path, and the only such.

To be narrowly specific: Might every regular polygon have a simple periodic path?

I know (or, rather, believe) that it remains unknown whether every polygon has a periodic billiard path. But Masur proved in the 1980's that every rational polygon (vertex angles rational multiples of $\pi$) has (many) periodic billiard paths.

---

      ![Hooper][1]
      ^{(Image from: W. Patrick Hooper, "Some irrational polygons have many periodic billiard paths." [PDF download link](http://wphooper.com/docs/talks/2008/milwaukee_printable.pdf).)}

---

My question concerns *simple* paths: non-self-intersecting, i.e., embedded paths:

Q. Is it known that certain classes of rational polygons have simple periodic billiard paths? If so, which? That certain classes (of rational polygons) have no such simple periodic paths? If so, which?

For example, in an acute triangle, its pedal triangle is a simple periodic path, and the only such.

I know (or, rather, believe) that it remains unknown whether every polygon has a periodic billiard path. But Masur proved in the 1980's that every rational polygon (vertex angles rational multiples of $\pi$) has (many) periodic billiard paths.

---

      ![Hooper][1]
      ^{(Image from: W. Patrick Hooper, "Some irrational polygons have many periodic billiard paths." [PDF download link](http://wphooper.com/docs/talks/2008/milwaukee_printable.pdf).)}

---

My question concerns *simple* paths: non-self-intersecting, i.e., embedded paths:

Q. Is it known that certain classes of rational polygons have simple periodic billiard paths? If so, which? That certain classes (of rational polygons) have no such simple periodic paths? If so, which?

For example, in an acute triangle, its pedal triangle is a simple periodic path, and the only such.

To be narrowly specific: Might every regular polygon have a simple periodic path?

I know (or, rather, believe) that it remains unknown whether every polygon has a periodic billiard path. But Mazur proved in the 1980's that every rational polygon (vertex angles rational multiples of $\pi$) has (many) periodic billiard paths.

---

      ![Hooper][1]
      ^{(Image from: W. Patrick Hooper, "Some irrational polygons have many periodic billiard paths." [PDF download link](http://wphooper.com/docs/talks/2008/milwaukee_printable.pdf).)}

---

My question concerns *simple* paths: non-self-intersecting, i.e., embedded paths:

Q. Is it known that certain classes of rational polygons have simple periodic billiard paths? If so, which? That certain classes (of rational polygons) have no such simple periodic paths? If so, which?

For example, in an acute triangle, its pedal triangle is a simple periodic path, and the only such.

I know (or, rather, believe) that it remains unknown whether every polygon has a periodic billiard path. But Masur proved in the 1980's that every rational polygon (vertex angles rational multiples of $\pi$) has (many) periodic billiard paths.

---

      ![Hooper][1]
      ^{(Image from: W. Patrick Hooper, "Some irrational polygons have many periodic billiard paths." [PDF download link](http://wphooper.com/docs/talks/2008/milwaukee_printable.pdf).)}

---

My question concerns *simple* paths: non-self-intersecting, i.e., embedded paths:

Q. Is it known that certain classes of rational polygons have simple periodic billiard paths? If so, which? That certain classes (of rational polygons) have no such simple periodic paths? If so, which?

For example, in an acute triangle, its pedal triangle is a simple periodic path, and the only such.