The trikite shown below can be used to form a new set of polyforms called Polyflaptiles.
The 14 one-sided tetrakites are shown below. The trepezia are made with one piece used twice.
Brendan Owen has produced a number of figures for this set.
Brendan also produced the following trapezium made with the 27 pentakites
and also found a solution to the following construction.
These figures were found using Peter Esser's trisolve program. The trapezium at the top right is the same as Brendan's except that it is divided into a parallelogram and a trapezium (Brendan has gone further and spilt this trapezium into three trapezia). It is possible to make three copies of this smaller trapezium with the set.
The trapezium above is an enlarged model of a pentiamond. Models of the other three can also be made.
Brendan has also made some other pentakite figures based on a hexagon.
There are 48 one-sided pentakites. This first diagram shows all possible paralleograms and trapezia which can be made with the set..
It is possible to make enlarged copies of each of the pentiamonds with this set.
Other symmetric constructions are possible. The figure at the left is an examples of a triangle with a hole in the shape of a trikite. There should be three other such constructions. The costruction at the top right consists of four congruent parts. Brendan Owen has shown that it is impossible to find eight hexagons (half of each of these shapes) but has found the two sets of eight consgruent shapes below.
Brendan Owen has packed the set of 44 pieces obtained by combining the kites from one to five into a triangle of side eight. Below are two pairs of parallelograms made with this set together with a hexagon made from the one-sided combined set.
Brendan has also made the figure below made with the 84 hexakites without an internal hole.
Patrick Hamlyn has improved on this by finding a solution made up of seven separate hexagons and has also made the two stars.The first star is made up of six congruent parts.
Peter Esser's solver also allows us to find solutions for chequered polykites.
In counting the perimeter of a polykite we can use the length of the longer side of the kite as our unit. On this basis there are nine perimeter 6 pieces with a total area of 39 units and 25 perimeter seven pieces with a total area of 135 units. The first diagram shows two constructions with the perimeter six set and a construction with the 15 pieces one-sided set.
