Free Egg of Life Grid Generator

Six equal circles touch a central one of the same size in a plane without overlapping, and no more. It is called the kissing number in two dimensions, and unlike its higher-dimensional cousins it is easy to see: six circles of radius r have centres 2r out, and those centres are 2r apart from each other, forming six equilateral triangles that close the full turn precisely.

The Egg, the Seed and the Fruit of Life get presented as a progression of complexity. They are more usefully understood as the same lattice at three different spacings, and the spacing is the entire difference.

The Seed of Life generator places seven circles with centres one radius apart, so every circle passes through its neighbours' centres and the figure is a mass of overlapping lenses. The Egg of Life places seven at exactly two radii, so they touch and nothing overlaps. The Fruit of Life places thirteen at a little over two radii, so they sit clear of each other and stay countable.

Three names, three spacings — r, 2r, and slightly more than 2r — and everything else about the figures follows. If you can hold that, the sequence stops needing to be memorised.

Start with circle packing. This arrangement is the kernel of the hexagonal packing, which is the densest possible packing of equal circles in a plane — a result proved rigorously in the twentieth century though known empirically for far longer. Anything that packs round things efficiently is looking at this figure repeated.

For making, the tangency is a construction convenience. Six tangent circles give you six contact points at exactly sixty degrees, which is a division of the circle you can then use for anything else — a bezel, a rosette, a set of holes. The contacts are unambiguous points, unlike the crossings of overlapping circles, which are pairs.

And for pattern, tangent circles tile. Step the whole seven-circle unit by 2r in any of the three lattice directions and the copies interlock exactly, with each outer circle of one unit becoming a shared member of the next. That is the move that turns the Egg of Life into a field.

The names in this sequence — Seed, Egg, Flower, Fruit — are twentieth-century and belong to a modern synthesis rather than to any older documented tradition. The underlying figures are genuinely ancient: overlapping-circle rosettes appear in Assyrian, Egyptian, Roman and medieval work, cut into stone and pressed into plaster across a very wide span.

What is not documented is that those older uses carried the meanings the modern names imply, or that they formed a sequence at all. The geometry is real and old; the narrative around it is recent. Drawing the figure because it is beautiful and useful needs no more than that.

One practical note on scale. Because the circles only touch, a printed or cut version has six single-point contacts, and a single point is exactly where a cutting path, a glaze or a printed line is least reliable. If the tangencies matter to the piece, check them at your actual output size — they are the first thing to break and the last thing you will notice on screen.

The seven-circle unit tiles, and the move is simple: step the whole figure by exactly 2r along any of the three lattice directions — sixty degrees apart — and the copies interlock. Each outer circle of one unit becomes a member of its neighbour's ring, which is why the field looks continuous rather than like a grid of separate flowers.

That is worth doing with the overlay's own transform rather than by duplicating an exported image, because the step distance has to be exactly 2r and a duplicated image placed by eye will drift. Place, step, place, step — checking each new position against the tangencies of the one before.

It is also the fastest way to understand why the hexagonal packing is the densest. After three or four steps the gaps between circles are visibly the smallest they could be, and the reason is directly in front of you: there is nowhere left to move a circle that does not open a bigger gap somewhere else.

One more thing the figure shows, and it is the reason the arrangement matters beyond ornament. Because six circles fit around one with no slack, the pattern is completely rigid: there is no way to rotate the ring, no way to slide one circle along without breaking contact somewhere, no free parameter at all. Circle packings in two dimensions are locked in a way sphere packings in three are not, and this seven-circle unit is the smallest place you can see that.

For anyone laying out physical work that is a practical guarantee rather than a curiosity. If your six outer positions are correct relative to the centre, they are correct relative to each other — you cannot get the ring right and the neighbours wrong. That is not true of most layouts, and it is why a hexagonal arrangement is forgiving to set out and unforgiving to fudge. The Egg of Life overlay page follows the sequence, beside the rest of the sacred-geometry overlays, and the Flower of Life generator draws where it ends up.