# IFS Attractors: Parallelograms

## Introduction

In some ways parallelograms are among the least interesting of self-similar tiles. A single construction generates all parallelograms as order 4 rep-tiles (which are Perron tiles) which tile the plane with a single copy in the unit cell, and one might consider this sufficient.

However there are 3 parallelograms which have lower order. The √2:1 rectangle is an order 2 Perron rep-tile, the √3:1 rectangle is an order 3 Perron rep-tile, and the golden rectangle is an order 3 Perron irrep-tile. The corresponding parallelograms are pseudo-Perron tiles.

All of these are the lowest order members of larger classes of contructions: the order 4 tiles of the rep-n2-parallograms, √2:1 rectangle and √3:1 rectangles of the rep-mn-parallelograms, and the golden tile of the metallic parallelograms.

There are many other ways of dissecting parallelograms into similar parallelograms. A subset of these, in which a square of integer side is dissected into smaller squares also of integer side have been investigated at Erich Friedman's Math Magic pages. (When generalised to parallelograms these these become ir-reptiles with IFS { p →ki/np + vi : ki, n ∈ ℤ}.) They lose the property that sides are of integer size (except from rhombi), but the ratio of areas remains the ratio of a pair of integer squares.)

## Rep-n2 Parallelograms

Any parallelogram is a rep-n2 rep-tile, and tiles the plane with one copy per unit cell.

Specific classes of parallelograms are squares (all angles and sides equal), rectangles (all angles equal) and rhombi (all sides equal). Several classes of polyforms include parallelograms, including polyominos, polyplets, polyabolos (aka polytans), polyiamonds and polydrafters.

### Examples

 square (monomino) rectangle (domino) rhombus (diamond) parallelogram (diabolo) parallelogram (didrafter)

### IFS

In the simplest construction of these figures all the transforms are n-fold contractions combined with translations. This may be written

Trs: p → p/n + ra + sb, r,s ∈0..n-1, where a and b are non-colinear vectors.

These IFSs have two degrees of freedom, which means that their attractors include an uncountably infinite number of dissimilar parallelograms. The degrees of freedom may be parameterised as stretch and skew. If the archetypal IFS is made that of the square with a = 1 and b = i, then the generalised IFS can be written as a = t and b = k + i, i.e.

Trs: p → p/n + rt + sk + si, r,s ∈0..n-1

### Degeneracy

However, because of the symmetry of the attractors, the attractors are degenerate, i.e. any particular parallelogram can be produced by more than one IFS. In all cases any element of the parallelogram may be rotated by 180°, giving 2n2 different IFSs. In the case of rectangles and rhombi any element may be reflected in either or both of two axes, giving 4n2 different IFSs. (Reflection in both axes is the same as rotation by 180°.) In the case of squares any element may also be rotated by 90° or 270°, instead of 0° or 180°, giving 8n2 different IFSs.

Furthermore some particular figures have additional IFSs; for example in the domino pairs of elements form a square, and any or all these can collectively by rotated by 90°. Thus the rep-4 domino is 16384 (4.84) rather than 4096 (84) -fold degenerate.

This degeneracy becomes significant when applying various algorithmic methods of generating new reptiles and irreptiles from a given reptile or irreptile.

## Rep-mn Rectangles

There are also rep-mn rectangles, including two with rep- numbers less than 4.

 rep-n rectangle √2:1 (rep-2) rectangle √3:1 (rep-3) rectangle 2:1 (rep-4) rectangle √5:1 (rep-5) rectangle √6:1 (rep-6) rectangle rep-2n rectangle (rep-4) square √3:√2 (rep-6) rectangle √2:1 (rep-8) rectangle √5:√2 (rep-10) rectangle √3:1 (rep-12) rectangle rep-3n rectangle √3:√2 (rep-6) rectangle (rep-9) square 2:√3 (rep-12) rectangle √5:√3 (rep-15) rectangle √2:1 (rep-18) rectangle rep-4n rectangle 2:1 (rep-8) rectangle 2:√3 (rep-12) rectangle (rep-16) square √5:√4 (rep-20) rectangle √3:√2 (rep-24) rectangle

The rep-nn rectangle is square (one of the degenerate alternatives to the rep-n2 square above). The rep-nm rectangle is the same as the rep-mn rectangle, but rotated by 90°.

### IFS

The simplest construction is a combination of an anticlockwise rotation by 90°, combined with contraction by √(mn) and a translation, which may be written as

Trs: p → peiπ/2/√(mn) + r1 + (s-1)√(m/n)i, r ∈1..n, s ∈1..m, where 1 and i are orthogonal unit vectors.

These IFSs have no degrees of freedom, their attractors being the √n:m rectangles. These can be placed in a one-to-one correspondence with the rational numbers (m/n), and hence form a countably infinite set of attractors.

### Degeneracy

Rep-mn rectangles are in general 4mn degenerate. Some specific instances (e.g. square and domino - see above) have greater degrees of degeneracy.

## Rep-mn Parallelograms

The √n:m rectangles can also be generated by a combination of reflection in the x-axis, anticlockwise rotation by 90°, contraction by √(mn) and a translation, which may be written as

Trs: p → p̄eiπ/2/√(mn) + (r-1)1 + (s-1)√(m/n)i, r ∈1..n, s ∈1..m, where 1 and i are orthogonal unit vectors.

 rep-n rectangle √2:1 (rep-2) rectangle √3:1 (rep-3) rectangle 2:1 (rep-4) rectangle √5:1 (rep-5) rectangle √6:1 (rep-6) rectangle rep-2n rectangle (rep-4) square √3:2 (rep-6) rectangle √2:1 (rep-8) rectangle √5:2 (rep-10) rectangle √3:1 (rep-12) rectangle rep-3n rectangle √3:2 (rep-6) rectangle (rep-9) square 3:2 (rep-12) rectangle √5:3 (rep-15) rectangle √6:3 (rep-18) rectangle

These are a special case of the rep-mn parallelograms, which are sets of attractors with a single degree of freedom, which can be parameterised as the angle of the lower left corner. The IFS transforms then become

Trs: p → p̄eω/2/√(mn) + (r-1)1 + (s-1)√(m/n)i, r ∈1..n, s ∈1..m, where 1 and i are orthogonal unit vectors.

rep-2 parallelogram rep-3 parallelogram rep-4 parallelogram rep-4 parallelogram rep-6 parallelogram

## Mixed Orientation Parallelograms

It is also possible to generate rectangles composed of alternating columns (or rows) of rotated and unrotated elements.

11121,1,1,2 1.4142135623731 11131,1,1,3 1.2247448713916 11141,1,1,4 1.1547005383793 11151,1,1,5 1.1180339887499 11161,1,1,6 1.0954451150103 11171,1,1,7 1.0801234497346 11181,1,1,8 1.0690449676497 11231,1,2,3 1.7320508075689 11241,1,2,4 1.4142135623731 11251,1,2,5 1.2909944487358 11261,1,2,6 1.2247448713916 11271,1,2,7 1.1832159566199 11281,1,2,8 1.1547005383793 11341,1,3,4 2 11351,1,3,5 1.5811388300842 11361,1,3,6 1.4142135623731 11371,1,3,7 1.3228756555323 11381,1,3,8 1.2649110640674 11451,1,4,5 2.2360679774998 11461,1,4,6 1.7320508075689 11471,1,4,7 1.5275252316519 11481,1,4,8 1.4142135623731 11561,1,5,6 2.4494897427832 11571,1,5,7 1.870828693387 11581,1,5,8 1.6329931618555 11671,1,6,7 2.6457513110646 11681,1,6,8 2 11781,1,7,8 2.8284271247462 12121,2,1,2 1 12131,2,1,3 0.86602540378444 12141,2,1,4 0.81649658092773 12151,2,1,5 0.79056941504209 12161,2,1,6 0.77459666924148 12171,2,1,7 0.76376261582597 12181,2,1,8 0.75592894601845 12231,2,2,3 1.2247448713916 12241,2,2,4 1 12251,2,2,5 0.91287092917528 12261,2,2,6 0.86602540378444 12271,2,2,7 0.83666002653408 12281,2,2,8 0.81649658092773 12341,2,3,4 1.4142135623731 12351,2,3,5 1.1180339887499 12361,2,3,6 1 12371,2,3,7 0.93541434669349 12381,2,3,8 0.89442719099992 12451,2,4,5 1.5811388300842 12461,2,4,6 1.2247448713916 12471,2,4,7 1.0801234497346 12481,2,4,8 1 12561,2,5,6 1.7320508075689 12571,2,5,7 1.3228756555323 12581,2,5,8 1.1547005383793 12671,2,6,7 1.870828693387 12681,2,6,8 1.4142135623731 12781,2,7,8 2 13121,3,1,2 0.81649658092773 13131,3,1,3 0.70710678118655 13141,3,1,4 0.66666666666667 13151,3,1,5 0.6454972243679 13161,3,1,6 0.63245553203368 13171,3,1,7 0.62360956446232 13181,3,1,8 0.61721339984837 13231,3,2,3 1 13241,3,2,4 0.81649658092773 13251,3,2,5 0.74535599249993 13261,3,2,6 0.70710678118655 13271,3,2,7 0.68313005106397 13281,3,2,8 0.66666666666667 13341,3,3,4 1.1547005383793 13351,3,3,5 0.91287092917528 13361,3,3,6 0.81649658092773 13371,3,3,7 0.76376261582597 13381,3,3,8 0.73029674334022 13451,3,4,5 1.2909944487358 13461,3,4,6 1 13471,3,4,7 0.8819171036882 13481,3,4,8 0.81649658092773 13561,3,5,6 1.4142135623731 13571,3,5,7 1.0801234497346 13581,3,5,8 0.94280904158206 13671,3,6,7 1.5275252316519 13681,3,6,8 1.1547005383793 13781,3,7,8 1.6329931618555 14121,4,1,2 0.70710678118655 14131,4,1,3 0.61237243569579 14141,4,1,4 0.57735026918963 14151,4,1,5 0.55901699437495 14161,4,1,6 0.54772255750517 14171,4,1,7 0.54006172486732 14181,4,1,8 0.53452248382485 14231,4,2,3 0.86602540378444 14241,4,2,4 0.70710678118655 14251,4,2,5 0.6454972243679 14261,4,2,6 0.61237243569579 14271,4,2,7 0.59160797830996 14281,4,2,8 0.57735026918963 14341,4,3,4 1 14351,4,3,5 0.79056941504209 14361,4,3,6 0.70710678118655 14371,4,3,7 0.66143782776615 14381,4,3,8 0.63245553203368 14451,4,4,5 1.1180339887499 14461,4,4,6 0.86602540378444 14471,4,4,7 0.76376261582597 14481,4,4,8 0.70710678118655 14561,4,5,6 1.2247448713916 14571,4,5,7 0.93541434669349 14581,4,5,8 0.81649658092773 14671,4,6,7 1.3228756555323 14681,4,6,8 1 14781,4,7,8 1.4142135623731 15121,5,1,2 0.63245553203368 15131,5,1,3 0.54772255750517 15141,5,1,4 0.51639777949432 15151,5,1,5 0.5 15161,5,1,6 0.48989794855664 15171,5,1,7 0.48304589153965 15181,5,1,8 0.47809144373376 15231,5,2,3 0.77459666924148 15241,5,2,4 0.63245553203368 15251,5,2,5 0.57735026918963 15261,5,2,6 0.54772255750517 15271,5,2,7 0.52915026221292 15281,5,2,8 0.51639777949432 15341,5,3,4 0.89442719099992 15351,5,3,5 0.70710678118655 15361,5,3,6 0.63245553203368 15371,5,3,7 0.59160797830996 15381,5,3,8 0.56568542494924 15451,5,4,5 1 15461,5,4,6 0.77459666924148 15471,5,4,7 0.68313005106397 15481,5,4,8 0.63245553203368 15561,5,5,6 1.0954451150103 15571,5,5,7 0.83666002653408 15581,5,5,8 0.73029674334022 15671,5,6,7 1.1832159566199 15681,5,6,8 0.89442719099992 15781,5,7,8 1.2649110640674 16121,6,1,2 0.57735026918963 16131,6,1,3 0.5 16141,6,1,4 0.47140452079103 16151,6,1,5 0.45643546458764 16161,6,1,6 0.44721359549996 16171,6,1,7 0.4409585518441 16181,6,1,8 0.43643578047198 16231,6,2,3 0.70710678118655 16241,6,2,4 0.57735026918963 16251,6,2,5 0.52704627669473 16261,6,2,6 0.5 16271,6,2,7 0.48304589153965 16281,6,2,8 0.47140452079103 16341,6,3,4 0.81649658092773 16351,6,3,5 0.6454972243679 16361,6,3,6 0.57735026918963 16371,6,3,7 0.54006172486732 16381,6,3,8 0.51639777949432 16451,6,4,5 0.91287092917528 16461,6,4,6 0.70710678118655 16471,6,4,7 0.62360956446232 16481,6,4,8 0.57735026918963 16561,6,5,6 1 16571,6,5,7 0.76376261582597 16581,6,5,8 0.66666666666667 16671,6,6,7 1.0801234497346 16681,6,6,8 0.81649658092773 16781,6,7,8 1.1547005383793 17121,7,1,2 0.53452248382485 17131,7,1,3 0.46291004988628 17141,7,1,4 0.43643578047198 17151,7,1,5 0.42257712736426 17161,7,1,6 0.41403933560541 17171,7,1,7 0.40824829046386 17181,7,1,8 0.40406101782088 17231,7,2,3 0.65465367070798 17241,7,2,4 0.53452248382485 17251,7,2,5 0.48795003647427 17261,7,2,6 0.46291004988628 17271,7,2,7 0.44721359549996 17281,7,2,8 0.43643578047198 17341,7,3,4 0.75592894601845 17351,7,3,5 0.5976143046672 17361,7,3,6 0.53452248382485 17371,7,3,7 0.5 17381,7,3,8 0.47809144373376 17451,7,4,5 0.84515425472852 17461,7,4,6 0.65465367070798 17471,7,4,7 0.57735026918963 17481,7,4,8 0.53452248382485 17561,7,5,6 0.92582009977255 17571,7,5,7 0.70710678118655 17581,7,5,8 0.61721339984837 17671,7,6,7 1 17681,7,6,8 0.75592894601845 17781,7,7,8 1.0690449676497 18121,8,1,2 0.5 18131,8,1,3 0.43301270189222 18141,8,1,4 0.40824829046386 18151,8,1,5 0.39528470752105 18161,8,1,6 0.38729833462074 18171,8,1,7 0.38188130791299 18181,8,1,8 0.37796447300923 18231,8,2,3 0.61237243569579 18241,8,2,4 0.5 18251,8,2,5 0.45643546458764 18261,8,2,6 0.43301270189222 18271,8,2,7 0.41833001326704 18281,8,2,8 0.40824829046386 18341,8,3,4 0.70710678118655 18351,8,3,5 0.55901699437495 18361,8,3,6 0.5 18371,8,3,7 0.46770717334674 18381,8,3,8 0.44721359549996 18451,8,4,5 0.79056941504209 18461,8,4,6 0.61237243569579 18471,8,4,7 0.54006172486732 18481,8,4,8 0.5 18561,8,5,6 0.86602540378444 18571,8,5,7 0.66143782776615 18581,8,5,8 0.57735026918963 18671,8,6,7 0.93541434669349 18681,8,6,8 0.70710678118655 18781,8,7,8 1 21122,1,1,2 2 21132,1,1,3 1.7320508075689 21142,1,1,4 1.6329931618555 21152,1,1,5 1.5811388300842 21162,1,1,6 1.549193338483 21172,1,1,7 1.5275252316519 21182,1,1,8 1.5118578920369 21232,1,2,3 2.4494897427832 21242,1,2,4 2 21252,1,2,5 1.8257418583506 21262,1,2,6 1.7320508075689 21272,1,2,7 1.6733200530682 21282,1,2,8 1.6329931618555 21342,1,3,4 2.8284271247462 21352,1,3,5 2.2360679774998 21362,1,3,6 2 21372,1,3,7 1.870828693387 21382,1,3,8 1.7888543819998 21452,1,4,5 3.1622776601684 21462,1,4,6 2.4494897427832 21472,1,4,7 2.1602468994693 21482,1,4,8 2 21562,1,5,6 3.4641016151378 21572,1,5,7 2.6457513110646 21582,1,5,8 2.3094010767585 21672,1,6,7 3.7416573867739 21682,1,6,8 2.8284271247462 21782,1,7,8 4 22122,2,1,2 1.4142135623731 22132,2,1,3 1.2247448713916 22142,2,1,4 1.1547005383793 22152,2,1,5 1.1180339887499 22162,2,1,6 1.0954451150103 22172,2,1,7 1.0801234497346 22182,2,1,8 1.0690449676497 22232,2,2,3 1.7320508075689 22242,2,2,4 1.4142135623731 22252,2,2,5 1.2909944487358 22262,2,2,6 1.2247448713916 22272,2,2,7 1.1832159566199 22282,2,2,8 1.1547005383793 22342,2,3,4 2 22352,2,3,5 1.5811388300842 22362,2,3,6 1.4142135623731 22372,2,3,7 1.3228756555323 22382,2,3,8 1.2649110640674 22452,2,4,5 2.2360679774998 22462,2,4,6 1.7320508075689 22472,2,4,7 1.5275252316519 22482,2,4,8 1.4142135623731 22562,2,5,6 2.4494897427832 22572,2,5,7 1.870828693387 22582,2,5,8 1.6329931618555 22672,2,6,7 2.6457513110646 22682,2,6,8 2 22782,2,7,8 2.8284271247462 23122,3,1,2 1.1547005383793 23132,3,1,3 1 23142,3,1,4 0.94280904158206 23152,3,1,5 0.91287092917528 23162,3,1,6 0.89442719099992 23172,3,1,7 0.8819171036882 23182,3,1,8 0.87287156094397 23232,3,2,3 1.4142135623731 23242,3,2,4 1.1547005383793 23252,3,2,5 1.0540925533895 23262,3,2,6 1 23272,3,2,7 0.9660917830793 23282,3,2,8 0.94280904158206 23342,3,3,4 1.6329931618555 23352,3,3,5 1.2909944487358 23362,3,3,6 1.1547005383793 23372,3,3,7 1.0801234497346 23382,3,3,8 1.0327955589886 23452,3,4,5 1.8257418583506 23462,3,4,6 1.4142135623731 23472,3,4,7 1.2472191289246 23482,3,4,8 1.1547005383793 23562,3,5,6 2 23572,3,5,7 1.5275252316519 23582,3,5,8 1.3333333333333 23672,3,6,7 2.1602468994693 23682,3,6,8 1.6329931618555 23782,3,7,8 2.3094010767585 24122,4,1,2 1 24132,4,1,3 0.86602540378444 24142,4,1,4 0.81649658092773 24152,4,1,5 0.79056941504209 24162,4,1,6 0.77459666924148 24172,4,1,7 0.76376261582597 24182,4,1,8 0.75592894601845 24232,4,2,3 1.2247448713916 24242,4,2,4 1 24252,4,2,5 0.91287092917528 24262,4,2,6 0.86602540378444 24272,4,2,7 0.83666002653408 24282,4,2,8 0.81649658092773 24342,4,3,4 1.4142135623731 24352,4,3,5 1.1180339887499 24362,4,3,6 1 24372,4,3,7 0.93541434669349 24382,4,3,8 0.89442719099992 24452,4,4,5 1.5811388300842 24462,4,4,6 1.2247448713916 24472,4,4,7 1.0801234497346 24482,4,4,8 1 24562,4,5,6 1.7320508075689 24572,4,5,7 1.3228756555323 24582,4,5,8 1.1547005383793 24672,4,6,7 1.870828693387 24682,4,6,8 1.4142135623731 24782,4,7,8 2 25122,5,1,2 0.89442719099992 25132,5,1,3 0.77459666924148 25142,5,1,4 0.73029674334022 25152,5,1,5 0.70710678118655 25162,5,1,6 0.69282032302755 25172,5,1,7 0.68313005106397 25182,5,1,8 0.67612340378281 25232,5,2,3 1.0954451150103 25242,5,2,4 0.89442719099992 25252,5,2,5 0.81649658092773 25262,5,2,6 0.77459666924148 25272,5,2,7 0.74833147735479 25282,5,2,8 0.73029674334022 25342,5,3,4 1.2649110640674 25352,5,3,5 1 25362,5,3,6 0.89442719099992 25372,5,3,7 0.83666002653408 25382,5,3,8 0.8 25452,5,4,5 1.4142135623731 25462,5,4,6 1.0954451150103 25472,5,4,7 0.9660917830793 25482,5,4,8 0.89442719099992 25562,5,5,6 1.549193338483 25572,5,5,7 1.1832159566199 25582,5,5,8 1.0327955589886 25672,5,6,7 1.6733200530682 25682,5,6,8 1.2649110640674 25782,5,7,8 1.7888543819998 26122,6,1,2 0.81649658092773 26132,6,1,3 0.70710678118655 26142,6,1,4 0.66666666666667 26152,6,1,5 0.6454972243679 26162,6,1,6 0.63245553203368 26172,6,1,7 0.62360956446232 26182,6,1,8 0.61721339984837 26232,6,2,3 1 26242,6,2,4 0.81649658092773 26252,6,2,5 0.74535599249993 26262,6,2,6 0.70710678118655 26272,6,2,7 0.68313005106397 26282,6,2,8 0.66666666666667 26342,6,3,4 1.1547005383793 26352,6,3,5 0.91287092917528 26362,6,3,6 0.81649658092773 26372,6,3,7 0.76376261582597 26382,6,3,8 0.73029674334022 26452,6,4,5 1.2909944487358 26462,6,4,6 1 26472,6,4,7 0.8819171036882 26482,6,4,8 0.81649658092773 26562,6,5,6 1.4142135623731 26572,6,5,7 1.0801234497346 26582,6,5,8 0.94280904158206 26672,6,6,7 1.5275252316519 26682,6,6,8 1.1547005383793 26782,6,7,8 1.6329931618555 27122,7,1,2 0.75592894601845 27132,7,1,3 0.65465367070798 27142,7,1,4 0.61721339984837 27152,7,1,5 0.5976143046672 27162,7,1,6 0.58554004376912 27172,7,1,7 0.57735026918963 27182,7,1,8 0.57142857142857 27232,7,2,3 0.92582009977255 27242,7,2,4 0.75592894601845 27252,7,2,5 0.69006555934235 27262,7,2,6 0.65465367070798 27272,7,2,7 0.63245553203368 27282,7,2,8 0.61721339984837 27342,7,3,4 1.0690449676497 27352,7,3,5 0.84515425472852 27362,7,3,6 0.75592894601845 27372,7,3,7 0.70710678118655 27382,7,3,8 0.67612340378281 27452,7,4,5 1.1952286093344 27462,7,4,6 0.92582009977255 27472,7,4,7 0.81649658092773 27482,7,4,8 0.75592894601845 27562,7,5,6 1.309307341416 27572,7,5,7 1 27582,7,5,8 0.87287156094397 27672,7,6,7 1.4142135623731 27682,7,6,8 1.0690449676497 27782,7,7,8 1.5118578920369 28122,8,1,2 0.70710678118655 28132,8,1,3 0.61237243569579 28142,8,1,4 0.57735026918963 28152,8,1,5 0.55901699437495 28162,8,1,6 0.54772255750517 28172,8,1,7 0.54006172486732 28182,8,1,8 0.53452248382485 28232,8,2,3 0.86602540378444 28242,8,2,4 0.70710678118655 28252,8,2,5 0.6454972243679 28262,8,2,6 0.61237243569579 28272,8,2,7 0.59160797830996 28282,8,2,8 0.57735026918963 28342,8,3,4 1 28352,8,3,5 0.79056941504209 28362,8,3,6 0.70710678118655 28372,8,3,7 0.66143782776615 28382,8,3,8 0.63245553203368 28452,8,4,5 1.1180339887499 28462,8,4,6 0.86602540378444 28472,8,4,7 0.76376261582597 28482,8,4,8 0.70710678118655 28562,8,5,6 1.2247448713916 28572,8,5,7 0.93541434669349 28582,8,5,8 0.81649658092773 28672,8,6,7 1.3228756555323 28682,8,6,8 1 28782,8,7,8 1.4142135623731 31123,1,1,2 2.4494897427832 31133,1,1,3 2.1213203435596 31143,1,1,4 2 31153,1,1,5 1.9364916731037 31163,1,1,6 1.897366596101 31173,1,1,7 1.870828693387 31183,1,1,8 1.8516401995451 31233,1,2,3 3 31243,1,2,4 2.4494897427832 31253,1,2,5 2.2360679774998 31263,1,2,6 2.1213203435596 31273,1,2,7 2.0493901531919 31283,1,2,8 2 31343,1,3,4 3.4641016151378 31353,1,3,5 2.7386127875258 31363,1,3,6 2.4494897427832 31373,1,3,7 2.2912878474779 31383,1,3,8 2.1908902300207 31453,1,4,5 3.8729833462074 31463,1,4,6 3 31473,1,4,7 2.6457513110646 31483,1,4,8 2.4494897427832 31563,1,5,6 4.2426406871193 31573,1,5,7 3.2403703492039 31583,1,5,8 2.8284271247462 31673,1,6,7 4.5825756949558 31683,1,6,8 3.4641016151378 31783,1,7,8 4.8989794855664 32123,2,1,2 1.7320508075689 32133,2,1,3 1.5 32143,2,1,4 1.4142135623731 32153,2,1,5 1.3693063937629 32163,2,1,6 1.3416407864999 32173,2,1,7 1.3228756555323 32183,2,1,8 1.309307341416 32233,2,2,3 2.1213203435596 32243,2,2,4 1.7320508075689 32253,2,2,5 1.5811388300842 32263,2,2,6 1.5 32273,2,2,7 1.4491376746189 32283,2,2,8 1.4142135623731 32343,2,3,4 2.4494897427832 32353,2,3,5 1.9364916731037 32363,2,3,6 1.7320508075689 32373,2,3,7 1.620185174602 32383,2,3,8 1.549193338483 32453,2,4,5 2.7386127875258 32463,2,4,6 2.1213203435596 32473,2,4,7 1.870828693387 32483,2,4,8 1.7320508075689 32563,2,5,6 3 32573,2,5,7 2.2912878474779 32583,2,5,8 2 32673,2,6,7 3.2403703492039 32683,2,6,8 2.4494897427832 32783,2,7,8 3.4641016151378 33123,3,1,2 1.4142135623731 33133,3,1,3 1.2247448713916 33143,3,1,4 1.1547005383793 33153,3,1,5 1.1180339887499 33163,3,1,6 1.0954451150103 33173,3,1,7 1.0801234497346 33183,3,1,8 1.0690449676497 33233,3,2,3 1.7320508075689 33243,3,2,4 1.4142135623731 33253,3,2,5 1.2909944487358 33263,3,2,6 1.2247448713916 33273,3,2,7 1.1832159566199 33283,3,2,8 1.1547005383793 33343,3,3,4 2 33353,3,3,5 1.5811388300842 33363,3,3,6 1.4142135623731 33373,3,3,7 1.3228756555323 33383,3,3,8 1.2649110640674 33453,3,4,5 2.2360679774998 33463,3,4,6 1.7320508075689 33473,3,4,7 1.5275252316519 33483,3,4,8 1.4142135623731 33563,3,5,6 2.4494897427832 33573,3,5,7 1.870828693387 33583,3,5,8 1.6329931618555 33673,3,6,7 2.6457513110646 33683,3,6,8 2 33783,3,7,8 2.8284271247462 34123,4,1,2 1.2247448713916 34133,4,1,3 1.0606601717798 34143,4,1,4 1 34153,4,1,5 0.96824583655185 34163,4,1,6 0.94868329805051 34173,4,1,7 0.93541434669349 34183,4,1,8 0.92582009977255 34233,4,2,3 1.5 34243,4,2,4 1.2247448713916 34253,4,2,5 1.1180339887499 34263,4,2,6 1.0606601717798 34273,4,2,7 1.024695076596 34283,4,2,8 1 34343,4,3,4 1.7320508075689 34353,4,3,5 1.3693063937629 34363,4,3,6 1.2247448713916 34373,4,3,7 1.145643923739 34383,4,3,8 1.0954451150103 34453,4,4,5 1.9364916731037 34463,4,4,6 1.5 34473,4,4,7 1.3228756555323 34483,4,4,8 1.2247448713916 34563,4,5,6 2.1213203435596 34573,4,5,7 1.620185174602 34583,4,5,8 1.4142135623731 34673,4,6,7 2.2912878474779 34683,4,6,8 1.7320508075689 34783,4,7,8 2.4494897427832 35123,5,1,2 1.0954451150103 35133,5,1,3 0.94868329805051 35143,5,1,4 0.89442719099992 35153,5,1,5 0.86602540378444 35163,5,1,6 0.84852813742386 35173,5,1,7 0.83666002653408 35183,5,1,8 0.82807867121083 35233,5,2,3 1.3416407864999 35243,5,2,4 1.0954451150103 35253,5,2,5 1 35263,5,2,6 0.94868329805051 35273,5,2,7 0.91651513899117 35283,5,2,8 0.89442719099992 35343,5,3,4 1.549193338483 35353,5,3,5 1.2247448713916 35363,5,3,6 1.0954451150103 35373,5,3,7 1.024695076596 35383,5,3,8 0.97979589711327 35453,5,4,5 1.7320508075689 35463,5,4,6 1.3416407864999 35473,5,4,7 1.1832159566199 35483,5,4,8 1.0954451150103 35563,5,5,6 1.897366596101 35573,5,5,7 1.4491376746189 35583,5,5,8 1.2649110640674 35673,5,6,7 2.0493901531919 35683,5,6,8 1.549193338483 35783,5,7,8 2.1908902300207 36123,6,1,2 1 36133,6,1,3 0.86602540378444 36143,6,1,4 0.81649658092773 36153,6,1,5 0.79056941504209 36163,6,1,6 0.77459666924148 36173,6,1,7 0.76376261582597 36183,6,1,8 0.75592894601845 36233,6,2,3 1.2247448713916 36243,6,2,4 1 36253,6,2,5 0.91287092917528 36263,6,2,6 0.86602540378444 36273,6,2,7 0.83666002653408 36283,6,2,8 0.81649658092773 36343,6,3,4 1.4142135623731 36353,6,3,5 1.1180339887499 36363,6,3,6 1 36373,6,3,7 0.93541434669349 36383,6,3,8 0.89442719099992 36453,6,4,5 1.5811388300842 36463,6,4,6 1.2247448713916 36473,6,4,7 1.0801234497346 36483,6,4,8 1 36563,6,5,6 1.7320508075689 36573,6,5,7 1.3228756555323 36583,6,5,8 1.1547005383793 36673,6,6,7 1.870828693387 36683,6,6,8 1.4142135623731 36783,6,7,8 2 37123,7,1,2 0.92582009977255 37133,7,1,3 0.80178372573727 37143,7,1,4 0.75592894601845 37153,7,1,5 0.7319250547114 37163,7,1,6 0.71713716560064 37173,7,1,7 0.70710678118655 37183,7,1,8 0.69985421222377 37233,7,2,3 1.1338934190277 37243,7,2,4 0.92582009977255 37253,7,2,5 0.84515425472852 37263,7,2,6 0.80178372573727 37273,7,2,7 0.77459666924148 37283,7,2,8 0.75592894601845 37343,7,3,4 1.309307341416 37353,7,3,5 1.0350983390135 37363,7,3,6 0.92582009977255 37373,7,3,7 0.86602540378444 37383,7,3,8 0.82807867121083 37453,7,4,5 1.4638501094228 37463,7,4,6 1.1338934190277 37473,7,4,7 1 37483,7,4,8 0.92582009977255 37563,7,5,6 1.6035674514745 37573,7,5,7 1.2247448713916 37583,7,5,8 1.0690449676497 37673,7,6,7 1.7320508075689 37683,7,6,8 1.309307341416 37783,7,7,8 1.8516401995451 38123,8,1,2 0.86602540378444 38133,8,1,3 0.75 38143,8,1,4 0.70710678118655 38153,8,1,5 0.68465319688146 38163,8,1,6 0.67082039324994 38173,8,1,7 0.66143782776615 38183,8,1,8 0.65465367070798 38233,8,2,3 1.0606601717798 38243,8,2,4 0.86602540378444 38253,8,2,5 0.79056941504209 38263,8,2,6 0.75 38273,8,2,7 0.72456883730947 38283,8,2,8 0.70710678118655 38343,8,3,4 1.2247448713916 38353,8,3,5 0.96824583655185 38363,8,3,6 0.86602540378444 38373,8,3,7 0.81009258730098 38383,8,3,8 0.77459666924148 38453,8,4,5 1.3693063937629 38463,8,4,6 1.0606601717798 38473,8,4,7