To calculate the 268 solutions of the Archimedes Stomachion is one thing, to investigate more forms the next. You easily find other symmetric forms, which have solutions with Stomachion tiles. We will use the simplified tiles and assume that all tiles fit on the base grid. Forms which do not conform to the base grid are indeed possible and will be discussed later.

After some playing with the tiles I put up the question, how many convex forms can be found, which have solutions with all Stomachion tiles. There are many more than I thought in the beginning and it was some work to find them all. The first approach is to construct all convex forms with area 144 and find out about solutions. I gave up because there are too many. So I added some constraints to discard forms, which for sure have no solution. Even after that move the number of forms is still about some ten thousand.

Some forms have many solutions, but there are others with a single one. The above forms have 197, 185, 48, 268 and 172 solutions.

The number and manifold of forms is amazing, compared to the small number of convex forms you get with squares and regular triangles. The ratio of solvable forms to possible forms get smaller the more vertices the form contains.

All forms with solutions a shown in the following pages, grouped by the vertex number. Forms appear only once, if they are symmetric. The maximum number of vertices is ten and there is only one possible form. It's my favorite Stomachion form and it is symmetrical of type S.

The following pages show a selected solution for each form with solutions. The total number of solutions for each form and other data are shown in the appended tables. I added also some forms without solutions.

 Number Pages Triangle 3 L 1-3 Tetragon 58 L 1-24   L 25-48   L 49-58   L X 1-29 Pentagon 104 L 1-26   L 27-52   L 53-79   L 80-104 Hexagon 198 L 1-21   L 22-45    L 46-69   L 70-93    L 94-119   L 12-144    L 145-167   L 168-198 Heptagon 181 L 1-24   L 25-50   L 51-75   L 76-100   L 101-127   L 128-155   L 156-181 Octagon 82 L 1-26   L 27-52   L 53-82 Nine and Ten Vertex 10+1 L 1-10+1

### Explanation for the tables on the solution pages:

Nr Current Number
Form
 The name coding of the form is defined by a sequence of segments. Each Segment defines the direction and the number of steps to the next vertex point. The direction names are according to the compass on the left. All segments are joined by a »+« symbol. Example: 12A+12D+24K means 12 steps into direction A, then 12 to D and finally 24 steps into direction K. This produces the triangle 3-1.
AP / IP
 Pick Theorem F = IP + AP / 2 - 1 ( = 144 ) IP denotes the number of inner and AP the number of outer grid points, i.e. lying on the edge of the form. Die values IP and AP are connected with the area F of the form. The beautiful Pick Theorem says how.
Sym
Type
Factor
Description

Q

8

symmetry of the square only occurs with the classical Stomachion

R

4

double symmetry of the rectangle, maybe diagonally

S

2

only rotation by 180°

M

2

mirror symmetry, horizontal, vertical or diagonal

W

2

only rotational symmetry

In the top examples the form 1 is of symmetry type »S«, although the geometry has two mirror axes! One of the mirrors is not conform to the grid and therefore useless.
The form 2 has no symmetry, 3 is of type »R« with two diagonal axis, 4 is of course mirror symmetric and 5 is of type »S«.

L

L is the number of solutions, if we dont count redundant symmetric solutions.

S

S is the number of backtracking steps needed to completely scan the search tree by the Logelium algorithm to find all solutions.

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