The problem asks for the maximal number of sides a convex polygon can have when part of a pair of convex polygons that tile the plane, under successively stronger conditions. The conditions, from weakest to strongest, are:

1. At least one copy of each tile.
2. At least ( k ) copies of each tile (for ( k \geq 2 )).
3. Infinitely many of each tile.
4. Every tile borders a tile of the other type...

Short answer For the two intermediate requirements,

• (3) there are infinitely many copies of each tile, and
• (4) every tile borders a tile of the other type, there is no finite upper bound on the number of sides: for every N ≥ 3 one can construct a plane tiling by two convex polygons in which one of the two tiles is an N‑gon. Thus the maximal number of sides is infinite for (3) and for (4). Unde...

The last two requirements that were still open in the question do ​not​ impose a finite upper bound.
For every integer N ≥ 3 one can make a tiling that

• uses exactly two different convex prototiles,
• contains infinitely many copies of each of them,
• makes every tile share at least one complete edge with a tile of the other kind,

and in which one of the tiles is an N-gon. Consequently...

Short answer.

Once we ask only that

(3) every shape occurs infinitely often, or even that

(4) in addition every tile has at least one edge in common with a tile of the other shape, there is still no upper bound at all on the number of sides that one of the polygons can have.
The first value that is really forced to be finite is obtained in the 2-isohedral case (condition 5) – there the ca...

Here is the state of affairs. Label your five “strength‐of‐hypothesis” requirements

(1) at least one copy of each polygon
(2) at least k copies of each
(3) infinitely many of each
(4) every tile shares an edge with at least one tile of the other type
(5) the tiling is 2–isohedral.

As you already observed

– under (1) or (2) one can do the usual “wedge” or “central‐star” cons...

<think> I need to figure out how many sides one of two convex polygons can have when they tile the plane together. The question has different conditions, from weakest to strongest, and I need to find the maximum number of sides for each condition.

First, recall that for a single convex polygon to tile the plane, it must be a triangle, quadrilateral, pentagon from one of 15 families, or a hexagon ...

This is an excellent and well-researched question. You've correctly identified the different levels of constraints and have already found the answer for the strongest case. Let's break down the other cases.

The crucial assumption in these problems is that the tiling is edge-to-edge. This means that for any two tiles that touch, they meet along a full edge of each. Your example construction fo...

Tiling the Plane with Two Convex Polygons: Maximum Number of Sides

This question explores how many sides a convex polygon can have when it's part of a pair of convex polygons that together tile the plane, under various restrictions.

What We Know

For single convex polygons that tile the plane, we have a complete classification:

• All triangles and quadrilaterals can tile the plane
• 15 fami...

This is a fascinating question about tilings of the plane by pairs of convex polygons. Let me analyze what we know about the bounds for each condition.

Known Results

For conditions 1 and 2 (at least 1 or k copies of each tile), you've correctly shown that the number of sides is unbounded using your wedge construction.

For condition 5 (2-isohedral tilings), you've found that 18 is the maximum...