Matched polygons

[bonanova]

Assuming a match is of unit length, it is possible to place 12 matches

on a plane in various ways to form polygons with integral areas.

I want to use the matches to make as many of these shapes as possible.

The entire match length must be used; mirror images and rotations do not count.

Examples:

A 3x3 square [with interior matches removed] uses 12 matches

with an area A = 9 square units. Another shape is a 1x5 rectangle,

with A = 5 square units.

You can attach two shapes; just remove the common edges, if any.

If you attach them only at a corner, there are no common edges.

Imagine a 2x2 square [8 matches] glued at a corner to a 1x1 square

[4 matches] to give a shape with 5 square units.

So far, I have found 30 shapes.

Their areas are varioulsy 9, 8, 7, 6, 5, 4 and 3 square units.

There may be more.

But now I've had a change of heart about gluing shapes together at a point.

Those are really shapes that use fewer than 12 matches, that just happen

to be touching. That eliminates seven polygons from my collection.

5 Square units: | is a vertical match, -- is a horizontal match, + is a match head, O is a touching point.

......+--+

......|..|

+--+--O--+

|.....|

+.....+

|.....|

+--+--+

4 square units

......+--+...+--+.........+--+--+

......|..|...|..|.........|.....|

+--+--O--+...+--O--+--+...+--+..+............+--+

|.....|.........|.....|......|..|............|..|

+--+..+.........+--+..+...+--O--+...+--+--+--O--+

...|..|............|..|...|..|......|........|

...+--+............+--+...+--+......+--+--+--+

3 square units

+--+

|..|

+--O--+.........+--+

...|..|.........|..|

...+--O--+...+--O--O--+

......|..|...|..|..|..|

......+--+...+--+..+--+

Can you find my other 23 shapes -- and possibly others?

Hint: Sort shapes by area.

[EventHorizon]

There's a large amount more....I've found 49 so far....and that's just in about a minute of looking.

If you constrain yourself to grid (unnecessary, I know) you can do something interesting. You can replace each of the squares with a rhombus. Lets say you have the 3x3 square with area 9. You can replace each square with a rhombus of area 8/9 to get a shape with 8 for its unit area. You can replace the squares with a rhombus of area 7/9, etc. So by "squishing" your square, I got 9 polygons...all unique to rotation and flips.

So for each rectilinear polygon you make, you actually get it's area added to the total number of shapes. Possibly more with shapes that are not symmetric....I'll think about it.

[EventHorizon]

I got what I think is your 23 rectilinear polygons, and by "squishing" (skewing) them I have found 138 unique polygons with integral sides. There is possibly more if I squish non-symmetric ones against "the wall" rather than "the floor" (still haven't thought much about that yet). I'll continue to look for more.

Having worked with pentominos and hexominos really helped with this.

[bonanova]

There's a large amount more....I've found 49 so far....and that's just in about a minute of looking.

Nice going.

When I made mine, I couldn't measure angles other than to make squares.

What was the smallest of your 23 "rectilinear" ones.?

Edited March 3, 2009 by bonanova
Simplify the instructions

[EventHorizon]

Nice going.

When I made mine, I couldn't measure angles other than to make squares.

What was the smallest of your 23 "rectilinear" ones.?

It was a pentomino (area of 5). But I've made another interesting discovery...

By skewing left instead of right on those shapes without a vertical or horizontal symmetry, I was able to get the number up to 210.

But I've found an even better way to get a lot of polygons.

Start with the 1x5 rectangle laying sideways. Choose an angle x and skew downwards the right 4 squares. Now you can skew the left (and only remaining square) enough to make the area integral (equal to the ceiling of the area of the four rhombuses on the right). You can see that these are all unique for 0<=x<90 by noticing that the two points that connect the two pieces remain where they were. Therefore....there are an infinite number of polygons with integral area values that can be created with 12 (infinitely thin) matches.

[bonanova]

EventHorizon points out that in some cases when a polygon whose interior comprises N contiguous unit-square areas, the angle between vertical and horizontal can be adjusted to make the N unit-square areas collectively decrease to N-1, N-2, N-3 ... 1 square units. The idea is that unit squares change into fractional rhombuses [rhombi?] that sum to an integer.

The shear can be done in different directions: left, right, upward or downward; and in some cases it can be done locally, leaving the remainder of the polygon unchanged. This increases dramatically the number of 12-match polygons with integral areas.

He shows further that when shear is performed independently on separate parts of a polygon -- but this must be done in a way that match count is preserved -- the separate groups need not both [all] have rational area values. That is, unit squares can be sheared in different parts of the polygon to acquire real [not a few rational] area values that sum to an integer. Thus an infinite number of integer-area polygons can be constructed using 12 matches.

Bravo.

For the purposes of stating the problem in simplest terms, and keeping the solution set finite, let us include shear groups with reflection and rotation symmetries. That is, if there is a class of polygons or classes of classes of polygons whose members have their corresponding vertices connected by edges of the same length, then count only the polygon in that group that has the largest area.

How many, now? And what size?