some tiles related to a cubic Pisot number

The polynomial x³ - 2x² + x - 1 has a single real root, r, with a value close to 1.754877666, which is a Pisot number. There are a number of IFS tiles in which the ratio of the areas of elements of dissections of the tile are a power of this number.

The simplest tiles which have this property are 3 element tiles, in which the areas of the 3 elements meet the constraint that a + a² + a⁴=1. This constraint is satisifed when a is the reciprocoal of the real root of the preceeding polynomial. The characteristic angle of rotation is w=99.343846°.

I have discovered four such tiles.

For the 1^st tile let T₁ be an c=a^½contraction combined with a rotation of w. Then the IFS is

• T₁
• T₂=T₁² + (1,0)
• T₃=T₁⁴ + (c³.cos(3w-p),c³.sin(3w-p))

For the 2^nd tile let T₁ be an c=a^½ contraction combined with a rotation of w+p. Then the IFS is

• T₁
• T₂=T₁² + (1,0)
• T₃=T₂² + rotation about (0.5,0) by p

For the 3^rd tile let T₁ be an c=a^½ contraction combined with a rotation of w-p. Then the IFS is

• T₁
• T₂=T₁² + (1,0)
• T₃=T₂.T₁² + (c.cos(w-p),c.sin(w-p))

For the 4^th tile let T₁ be an c=a^½ contraction combined with a rotation of w. Then the IFS is

• T₁
• T₂=T₁² + rotation about (0,0) by p + (1,0)
• T₃=T₁⁴ + (c.cos(w-p),c.sin(w-p))

All these figures tile the plane. (Images for the 3^rd and 4^th are yet to be produced. I note that the angle between the lattice vectors of the 3rd - w-p.)

In the first case the unit cell contains two copies of different sizes and orientations. The smaller copy is obtained from the large by the transform

• T=T₁ + (1+c.cos(w),c.sin(w))

The lattice vectors are

• (r.cos(p-2w),r.sin(p-2w)
• (r^½.cos(p-w),r^½.sin(p-w)

In the second case the unit cell contains two copies of the same size but different orientations. One copy is obtained from the other by the transform

• T=rotation by p + (r.cos(p-2w),r.sin(p-2w))

The lattice vectors are

• (r^1.5.cos(-3w),r^1.5.sin(-3w))
• (r^1.25.cos(a), r^1.25.cos(a))

The value of a is derived as follows. A triangle formed by the unit vectors and their difference has sides r^1.5, r^1.25 and r^0.75. From this the angle between the unit vectors can be obtained by application of the cosine rule. Subtracting this from -3w gives a, which is approximately 21.640°.

The following 5 element tiles can be derived from the above.

In all of these is can be observed that two copies make up the corresponding 3 element tile, and hence we can deduce that these figures tile the plane with 4 copies in the unit cell.

In addition 7 element tiles can also be produced from the original 3 tiles.

The same ratios and rotations also satisfy the constraint a + 2a³+ a⁵=1. I have found one 4 element tile corresponding to this constraint.

From this three 7-element tiles are immediately derived. (A 4^th possibility is not connected.)

Source: Independent discovery.

The constant in the Douady's Rabbit Fractal, a Julia Fractal, is close to, if not identical to, c.ei^p-w. Prior to my discovery of a procedure for ascertaining the characteristic rotations for tiles corresponding to unit cubic Pisot numbers with complex conjugates, a posting to news:sci.fractals by Roger Bagula brought p-w to my attention as a potential rotation value for tiles corresponding to the polynomialx³ - 2x² + x - 1, and led to my discovery of the 2^nd tile shown on this page. Further study identified a figure related to his claimed tile as being the unit cell for the 3^rd tile shown on this page, and led me to the discovery of that 3^rd tile.

© 2002 Stewart R. Hinsley