The tilings under the stars.

A star pattern is not drawn; it is grown on a net. Beneath every field of stars in this tradition lies a tessellation of polygons in contact, and the pattern is what happens when each polygon is given its star and the stars are made to meet. Which nets will do is a question with a short and beautiful answer.

The classical method builds a star pattern in two stages: first a tessellation of polygons that touch edge to edge, then a star in each polygon, drawn from its edge midpoints. This is the polygons-in-contact method that Ernest Hankin described from the buildings of India a century ago, and it is the method the studio's engine implements. The stars are the visible part. The tessellation is the part that decides everything, and it is worth knowing which tessellations there are.

Regular, then semi-regular

Start with the strictest case: one kind of regular polygon, meeting edge to edge, the same arrangement at every corner. There are exactly three. Triangles, six to a corner; squares, four to a corner; hexagons, three to a corner. Written by the polygons that meet at each vertex, they are 3.3.3.3.3.3, 4.4.4.4 and 6.6.6, and no other regular polygon tiles on its own, because the corner angles of a pentagon, heptagon or octagon do not divide 360.

Allow more than one kind of regular polygon, still meeting edge to edge and still with the same arrangement at every vertex, and the count rises to eight more. These are the Archimedean or semi-regular tilings: 3.12.12, two dodecagons and a triangle at each corner; 4.8.8, two octagons and a square; 4.6.12; 3.4.6.4; 3.6.3.6; 3.3.4.3.4; 3.3.3.4.4; and 3.3.3.3.6. Eleven tilings in all, three regular and eight semi-regular, a list that has been complete since Kepler wrote it down in 1619. The last of the eight, the snub hexagonal, is the odd one out: it comes in a left-handed and a right-handed form, and it is the only one of the eleven you can tell from its mirror image.

The nets the tradition chose

A star pattern wants a net of large polygons, because a star of n points needs an n-gon to sit in, and it wants those polygons in contact, so that the star lines leaving each edge meet the star lines from the neighbour. Run down the list with that in mind and the tradition's choices explain themselves.

• 6.6.6, the hexagonal net, carries the six-fold family. A six-star in every hexagon, meeting its six neighbours along six edges. It is the studio's default for six-fold work.
• 4.8.8, the octagon-and-square net, carries the eight-fold family and is the engine's default for it. An eight-star in every octagon, and in each square a four-armed cross: this is the star-and-cross of Fez and Granada, and the reason that pattern has a cross at all. The square is not decoration; it is the gap the octagons leave.
• 3.12.12, the dodecagon-and-triangle net, carries the twelve-fold family and is the default for it. A twelve-star in every dodecagon and a small three-fold motif in each triangle.
• 3.6.3.6, the kagome, hexagons and triangles alternating round each vertex, is the basket-weave net of Japanese bamboo and of the Mughal jaali. The studio offers it as an explicit seating for six- and twelve-fold stars, where the triangles between the stars sharpen a field that would otherwise read as discs.

What is missing from that list is as telling as what is on it. There is no decagon in any of the eleven, because a decagon's corner angle of 144 degrees fits no regular companion. So the ten-fold family, the richest in the tradition, is built on a net that is not Archimedean at all: decagons in contact on a rhombic lattice, with an irregular six-sided bowtie filling each gap. The tradition kept the polygons-in-contact rule and gave up the regularity of the filler, and the crystallographic restriction is the reason it had to.

Three sizes of star from one net

The Archimedean net with the most kinds of polygon, 4.6.12, has dodecagons, hexagons and squares meeting at every corner, and a pattern grown on it carries three sizes of star at once: twelve-pointed, six-pointed and four-pointed, in a fixed proportion the net dictates. Patterns of this kind are among the grandest in the Anatolian and Persian repertoire, and they show the method at its clearest. Nobody chose to put a six-star next to a twelve-star in that ratio. The net put the hexagon next to the dodecagon, and the stars followed.

Why the net is what gets cut

In a pierced screen the distinction between net and pattern disappears, because the net itself becomes the object. A Mughal jaali on the 4.8.8 net is a slab of sandstone with the star lines cut clean through it, and what holds the screen together is the geometry of the tessellation underneath: every bar meets its neighbours at a vertex of the net, so the load has somewhere to go. Draw the same field as line only and you are looking at the cutting plan.

Why the studio starts from the net

Because the net is where the exactness lives. Every vertex of a star pattern in the studio is the intersection of two lines that leave two polygon edges at a derived angle, and every one of those edges belongs to a tessellation the engine has laid out to the vertex. Change the net and the pattern changes with it, correctly: seat eight-stars on the square net instead of 4.8.8 and the crosses vanish, because there are no gaps to put them in. That is not an effect. It is the geometry doing what the geometry does, and it is the difference between a pattern that is derived from a tessellation and a picture that resembles one.