# Plants of Southern California:Pinus jeffreyi and P. ponderosa Cone Scale Arrangement

 Fig. 1. The top side of a ponderosa pine cone (note the dark color of the inner portion of the scales). Left: the 8 row low-angle spiral pattern. Right: the 13 row steeper-angle spiral pattern. This cone is the second from the right in the top row in this picture of the cones from the side. Click on the pictures for larger versions.

Introduction
Illustration of the Cone Scale Arrangement
Properties of the Cone Scale Arrangement
The Mathematics Behind the Cone Scale Arrangement

## Introduction

Pine cones, sunflower heads, and pineapples are three examples of beautiful spiral patterns in nature. It borders on being amazing that such a pattern could actually exist, when you think about what has to happen to create and keep that pattern as the parts grow. As you might imagine, you can't arbitrarily pick the arrangement of the segments in these examples, nor can you arbitrarily pick the number of segments. Only certain numbers of segments will work, and they have to fit together in a certain way.

For pine cones of these two species, it turns out that the pattern of the scales in the cones is quite interesting. The scales are in two interweaving rows. If you look at the cone from the side, one row has a shallow angle of about 30°, and there are either five or eight rows. The other row has a steeper angle of about 45°, and there are either eight or 13 rows.

I became interested in this pattern because the two row numbers have been used to separate Pinus jeffreyi and P. ponderosa in several references. For example, the paper by Haller and Vivrette 2011, which defined var. pacifica of P. ponderosa, stated that var. pacifica has row numbers of 5 and 8, whereas P. jeffreyi has row numbers of 8 and 13.

On 5 October 2014, Adrienne Ballwey and I set off on a pine survey in the San Bernardino Mountains. The major goal of this trip was to understand what was going on with the unusually-small Jeffrey cones we had seen on the PCT near Onyx Summit two months earlier. It turns out that Jeffrey cones produced in 2014 were unusually small due to the very low rainfall in 2013-2014.

We collected cones from a number of our stops, specifically to analyze the cone scale arrangement later at home.

For each collection at each location, the species determination was made using the combination of a histogram of cone sizes both from this year's cones and last year's cones; the needle color; the color difference of the two sides of the inner part of the cone scales; the bark chip inner color; the odor of the bark crevices; the vegetative buds (when available); and the appearance of the cone scale tip prickle. The determination of each collection was never ambiguous. In particular, every location that had very-small Jeffrey cones from this year, had mixed in with them normally-large Jeffrey cones from previous years.

Analysis of the cone scale arrangement of the collected cones, all of which are shown in Fig. 4 below, shows that the row numbers are not species dependent. Larger ponderosa cones have row numbers of 8 and 13, just like normal Jeffrey cones, and small Jeffrey cones have row numbers of 5 and 8, just like smaller ponderosa cones.

Fig. 1 shows an average size ponderosa cone in our collection that has row numbers of 8 and 13. Fig. 2 below shows a small Jeffrey cone from our collection that has row numbers of 5 and 8. The row number is almost surely simply a function of the size of the cone, and is not species dependent except in the sense that Jeffrey usually has larger cones than ponderosa.

## Illustration of the Cone Scale Arrangement

Fig. 2 shows a single small P. jeffreyi cone that has rows of 5 and 8. The cone has been rotated by something like 50° in each successive image, with the rows marked with numbered colored lines, so that you can see how the rows spiral around the cone. Since it is easier to see the rows in a closed cone, I submersed the open cone I collected in water until it closed. (See pictures of open cones and the same cones after immersion in water. The cone shown in this section is the rightmost Jeffrey cone in the top row in these pictures.)

 a b c d e f

Fig. 2. Illustration of the number of rows of scales in a pine cone, for one of the two sets of rows in the pattern, the one spiraling up to the right in these pictures.
The pine cones were rotated by roughly 50 degrees between successive pictures, from left to right in first the top row and then the second row, with the closest part of the cone moving to the left. The eight rows of scales spiraling up to the right in the pictures are marked with numbered colored lines, using five colors, with three colors repeated.
The rotation can be tracked by the two scales at the top in the pictures that are separating from the body of the cone, and in some pictures by features such as the brownish pit seen at middle right of (b), and at the left edge in (c). Many of the prickles are also distinctive enough to track in successive images.
Click on the pictures for larger versions.

Note that all eight rows end with a scale at the broad end of the cone, with row #8 next to row #1.

 a b c d e f

Fig. 3. Illustration of the number of rows of scales in a pine cone, for the other set of rows in the pattern, the one spiraling up to the left in these pictures.
The base pictures are the same as in Fig. 2. The pine cones were rotated by roughly 50 degrees between successive pictures, from left to right in first the top row and then the second row, with the closest part of the cone moving to the left. The rows of scales spiraling up to the right in the pictures are marked with numbered colored lines, using five colors.
The rotation can be tracked by the two scales at the top in the pictures that are separating from the body of the cone, and in some pictures by features such as the brownish pit seen at middle right of (b), and at the left edge in (c). Many of the prickles are also distinctive enough to track in successive images.
Click on the pictures for larger versions.

Note that all five rows end with a scale at the broad end of the cone, with row #5 next to row #1.

It seems like an apparent contradiction that both the eight row pattern and the five row pattern both end in scales at the broad end of the cone. The contradiction is resolved by noting that the scales at the broad end of the cone are not at the same horizontal level. For example, in Fig. 2e, the scales which terminate rows 7 and 8 at the broad end of the cone are part of the same row 3 in Fig. 3e. I.e., the rows with the steeper angle are more numerous because more of the uppermost scales participate in those rows. Another view of the ends of the rows, and how the ends are staggered near the wide part of the cone, in Fig. 1.

Note that the angle of the eight-row set is about 45°, and the angle of the five-row set is about 30°.

## Properties of the Cone Scale Arrangement

The first question is: do the cone scale spirals always have the same sense for each species?

Fig. 4 shows the angle of the rows that spiral up to the right, in all collected cones.

Fig. 4. Angle of the rows that spiral up to the right in this picture of cones collected over a large geographic and elevational range in the San Bernardino Mountains. Jeffrey cones are on the left (note their more ovate shape compared to the ponderosa cones); ponderosa cones on the right (note the non-tip portion of the scales is much darker, generally black, not chocolate-brown, on the side toward the broad end of the cone).
These are not a random sample of Jeffrey cones; I intentionally collected more of the very-small cones produced this year by some trees, since they were the focus of my study.
Click on the picture for a larger version.

Excluding the one ponderosa cone that appears to have two different angles spiraling up to the right, seven cones have an angle of 30 to 35° and six cones have an angle of 40 to 45°. These are roughly the two angles seen in the two different set of rows in the example above. Furthermore, both species have cones in both sets. This implies that there is no preference for the cones to have their spirals in a fixed sense for either species.

Some of the ponderosa cones in Fig. 4 have an 8-5 row pattern, as given for P. ponderosa var. pacifica in Haller and Vivrette 2011. However, some of the cones, such as the one shown in Fig. 1, have a 13-8 row pattern. The larger Jeffrey cones in Fig. 3 have a 13-8 row pattern, but the smaller cones have an 8-5 pattern. This is an extension of the information in Haller and Vivrette, who state that only Jeffrey has a 13-8 row pattern, and only ponderosa has a 8-5 row pattern.

Brousseau did a much-more thorough study of the patterns in Jeffrey and ponderosa cones, using a sample of 400 cones from both (see On the Trail of the California Pine and Fibonacci Statistics in Conifers). He found that both Jeffrey and ponderosa pine had cones with the following row numbers: 8-5; 13-5; and 13-8; unfortunately, he gave no information as to how abundant each pattern was. (Important caveat: there is no guarantee that he correctly distinguished these species.) He also found that most individual trees produced equal amount of cones spiraling in each direction.

The key in the Flora of North America Pinus treatment states that Jeffreyi has 8 or more scales per row, as viewed from side, in low spirals, whereas ponderosa has 5-7 in steep spirals. In the low spirals in Fig. 3, the Jeffrey cones have 5-8 scales per row, and the ponderosa cones have 6-8 scales per row in their steep spirals. Either I don't understand exactly what this key characteristic means, or it is not correct for the southern California species.

## The Mathematics Behind the Cone Scale Arrangement

The botanical term for the arrangement of plant parts is phyllotaxy, literally "leaf arrangement". The cone in Figs. 2 and 3 has "seed cone phyllotaxy of 5 rows in one direction, and 8 rows in the other". The cone in Fig. 1 has 8 rows in one direction, and 13 rows in the other.

The arrangement of the segments in these examples, and many other examples in nature such as a Nautilis shell, is in a logarithmic spiral (or helix), which has the advantage that the spiral always has the same shape no matter how large it is, or how many parts it has. You can imagine it was natural for plants and animals to evolve to produce such a shape, since it allows a good pattern to grow with time in the same way.

It turns out that the "certain numbers" in a logarithmic spiral are Fibonacci numbers, named after Leonardo of Pisa, the Italian mathematician who first discovered this sequence, who for some less-than-clear reason is now known as Fibonacci. He published this sequence in the year 1202, introducing it, and Arabic numerals, to Europe. The Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13, 21, ..., in which each number is the sum of the previous two numbers. Fibonacci numbers occur in nature in other ways than the logarithmic spiral as well.

It makes sense that Fibonacci numbers might appear in plants, since to grow larger, they must add a certain number of parts, and adding the previous number of parts to their current number of parts is a natural number to add.

As shown above, pine cones arrange their scales in something close to a logarithmic spiral in two directions. This arrangement always has two rows of parts, with rows in one direction being a given Fibonacci number, and rows in the other direction usually being the neighboring Fibonacci numbers.

I thank Rudi Schmid for alerting me to the paper by Haller and Vivrette 2011, which finally enlightened me about the phyllotaxy of the pine cones of these two species and stimulated this investigation. I thank Adrienne Ballwey for invaluable help in the field to collect the data and the cones, and make the determinations. I thank Jane Strong for comments that improved this webpage.

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