Ammonites: every possible shell shape, and why some were probably never built

How three numbers make an ammonite#

An ammonite grew by adding material only at the rim of its opening, so each new stretch of tube was, roughly, a slightly larger copy of the one before. Growth like that leaves a logarithmic spiral behind it. Many species still shifted their proportions a little as they grew.

In 1966 and 1967 the American palaeontologist David Raup showed that a few numbers are enough to draw almost any coiled shell of this kind. For ammonites, the page uses three:

• How much the tube grows each whorl (Raup called it W). At ×2, the tube is twice as big after one full turn.
• How far the tube stays from the centre (D): its inner edge divided by its outer edge. At 0 the tube touches the axis; at 0.6 a wide hole, the "navel", stays open in the middle.
• The shape of the cross-section (S): the tube's width divided by its height. The lower it is, the flatter the disc; the higher, the fatter the shell, until it is almost a ball.

Raup had a fourth number too, how far the spiral climbs along its axis, which turns the tube into a snail. The ammonites on this page all grew in a single plane, so for them it is zero.

Where the 201 real species sit#

The dots on the map are 201 ammonite species from the Triassic, Jurassic and Cretaceous, periods that span roughly 252 to 66 million years ago. Horacio Parent, Andrés Greco and Matías Bejas of the National University of Rosario, Argentina, gathered them from 1,003 specimens, measured in their lab or taken from the literature. For each species they published the smallest and largest diameter measured and the averages of the classic proportions: whorl height, whorl width and the opening of the navel, each relative to the diameter. Two years later Parent, Bejas and Greco, with Øyvind Hammer of the University of Oslo, gave the formulas that turn those proportions into Raup's numbers. Every dot on the map is calculated with them.

The species do not fill the map. All of them sit between ×1.44 and ×4.00 per whorl, and half grow more slowly than ×2.06. In other words, a typical ammonite doubled its diameter with every whorl, as the authors also noted.

The thin line on the map divides two worlds. Below it, each new whorl rests on the one before and partly covers it: 173 of the 201 species grew like that. On the line, the whorls just touch: 2 species. Above it, a gap stays open between whorls and the shell unwinds like a spring: 26 species. 20 of them are from the Early Cretaceous, 9 of those of the genus Crioceratites; the other 6 are scattered from the Late Triassic to the Late Cretaceous, among them the Middle Jurassic Spiroceras. The count uses the authors' own measure of overlap; on the map, where dots are drawn from the width of the navel, a few species fall a hair to the other side of the line.

Why some places stayed empty: size#

Most of the map is empty. Shells that widen more than four times per whorl are easy to draw, but no ammonite in the sample grew like that. Parent, Greco and Bejas offer a simple explanation: size.

Every ammonite started from a tiny first shell, the embryonic one. From published data, the authors take it to measure between 0.2 and 1.05 mm, and an adult to add between 5 and 9 whorls after it. If the tube widens W times per whorl, after N whorls the diameter is the starting one multiplied by W, N times over. In a tube that widens fast, a few whorls lead to enormous sizes.

The authors' example: a shell that widened ×12.8 per whorl would reach 67.5 metres across, starting from the smallest embryonic shell and after only five whorls. At the other end, a shell that widens by only 4% per whorl would stay at 1.5 mm, even after nine whorls from the largest embryonic shell. According to the authors, the adults in the sample measure between 10 and 495 mm.

The threshold shows on the map: above ×5.5 per whorl, even the smallest possible ammonite would pass a metre. Every real species sits well below it.

The idea can be tested with data from the same table. For each species, the range of sizes its W would allow can be calculated and compared with the largest diameter measured. That diameter falls inside the range for 189 of 201 species. The other 12 fall outside it, 7 above and 5 below: Paracadoceras efimovi, Gravesia gravesiana, Discotropites smithi, Acrioceras ornatum alpinum, Protacrioceras ornatum, Dactylioceras clevelandicum, Catutosphinctes araucanensis, Spiroceras annulatum, Emileia aff. dundriensis, Polyptychites stubendorffi, Paravascoceras carteri, Taramelliceras richei. Some changed shape as they grew, while the rule assumes the same W from the first whorl to the last. For some of the ones that come out too small, only young shells were measured; the ones that come out too big grew past what nine whorls from a 1.05 mm start allow, and the authors note that the number of whorls need not be constant, even within a single species.

The explanation remains a hypothesis. Of their proposed order of causes the authors write that it "is not the only which could be conceived, but in our view it is the most plausible". Their table gathers data from many sources and, they also write, is not suitable for statistical inference. What the map does show clearly is this: the shapes furthest from the real ones are those that would have produced ammonites the size of a building, or of a grain of sand.

One of the 201, from the Yorkshire coast#

In Whitby, ammonites are snakestones. According to the Natural History Museum, the legend dates to the seventh century and the story of the Saxon abbess Saint Hilda; "enterprising Victorians carved snake heads on the ammonite fossils", and three ammonites with snake heads became the device on the town's coat of arms.

One of the 201 species on the map comes from that coast: Dactylioceras clevelandicum, from Michael Howarth's 1973 study of the Upper Lias grey shales of the Yorkshire coast. It is an evolute, snake-like coil whose whorls barely overlap, with a wide navel, and it is also one of the 12 species whose real size falls outside the range the size rule allows: the largest shell measured is 81 mm, and the rule stops at 64 mm.

All 201 species, one by one#

Any row can be opened in the shell at the top. The numbers are calculated from the averages Parent and colleagues published; the diameters are the smallest and largest measured. Species names are spelled as in the source table.

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How it was calculated#

The data come from Appendix 1 of Parent, Greco and Bejas (2010): 201 species, averaged over 1,222 sets of measurements. Raup's numbers are calculated with equations 6 and 7 of Parent, Bejas, Greco and Hammer (2012): W = (1 − H2/D)⁻², where H2/D is the product of the H1/D and H2/H1 columns, and Raup's D is the navel diameter divided by the shell diameter. The section shape is whorl width divided by whorl height; for 21 species the width is not given in the appendix and the shell is drawn with the typical value. One row (Choffattia aff. neumayri) prints its diameters in reverse order in the source, 82–31; the table above shows them in order.

Adult size uses the relation in the 2010 paper: the diameter is that of the embryonic shell multiplied by W to the power of the number of whorls, with the ranges the authors adopted (0.2–1.05 mm and 5–9 whorls). The 67.5 m and 1.5 mm examples reproduce their calculations. The page's test compares, for each species, the largest diameter measured with the range the rule allows. "Closest species" compares growth per whorl, on a log scale, and distance from the centre, on the same scale as the map.

The shell drawn is Raup's geometric model, without the ribs, knobs or suture patterns that differ from species to species. Its colour is the page's choice, not a measurement.

Sources#

• Parent, H., Greco, A.F. and Bejas, M. (2010). Size–shape relationships in the Mesozoic planispiral ammonites. Acta Palaeontologica Polonica 55(1): 85–98. doi.org/10.4202/app.2009.0066
• Parent, H., Bejas, M., Greco, A. and Hammer, Ø. (2012). Relationships between dimensionless models of ammonoid shell morphology. Acta Palaeontologica Polonica 57(2): 445–447. doi.org/10.4202/app.2010.0118
• Natural History Museum, London. Snakestones: the myth, magic and science of ammonites. www.nhm.ac.uk/discover/snakestones-ammonites-myth-magic-science.html
• Raup, D.M. (1966). Geometric analysis of shell coiling: general problems. Journal of Paleontology 40(5): 1178–1190.
• Raup, D.M. (1967). Geometric analysis of shell coiling: coiling in ammonoids. Journal of Paleontology 41(1): 43–65.