Sunflower math

A golden rule of biology is that individuals attempt to leave as many offspring (i.e., genes) as possible in the next generation.  Plants, like the sunflowers growing in my backyard right now, try to maximize their seed production by packing seeds into the flower head in the most optimal way.  And that’s where the math comes in.

The sunflowers have grown quite tall (the fence posts are five feet), and some of the flowers are dinner-plate size (that’s a bumblebee on the flower head below for size reference).

But have you ever really inspected the interior of one of these complex flower heads?

The outer, yellow petals are really individual infertile ray flowers.  The center, disk flowers open from outer toward inner rows, a few layers at a time.  The yellow-tipped anthers stick up from the central part of each flower, presenting a disk of pollen for the lucky pollinators.

Once fertilized by the many bees that visit these flowers, each of the individual flowers produces one seed.  The entire flower head can produce as many seeds as there are individual disk flowers.  So the question is, how to pack as many of those flowers as possible into the circular head.

This pattern is not random.  Each disk flower (or potential seed) lies at an angle of 137.5 degrees from its neighbor.  Lined up from center to outer rim, the flowers (or seeds) describe two sets of spirals, one curving to the left, and one curving to the right.  This is the formula for maximal packing into circular space, and has been termed the “golden angle”.

As explained by mathematicians, divide a circle into two sections, such that the ratio of the large arc to the small arc is the same as the ratio of the entire circle to the large arc.  The golden angle is that created by the smaller of the two arcs, exactly 137.5 degrees.

(http://en.wikipedia.org/wiki/Golden_angle)

Applying the golden angle to the construction of the flowerhead, we get the following sort of pattern.

Pretty darn smart of those plants!

A crop of sunflowers growing in northwestern Minnesota near Crookston.

Lake Agassiz Beach

No, it’s not a new retirement hot spot.  The Lake Agassiz beach ridges are the only topographic relief (elevation) in the flattest country I’ve ever seen, in the northwestern corner of Minnesota.  They grow a lot of wheat here, as well as corn, soybeans, sugar beets, and sunflowers.  The wheat fields are miles long and half miles wide.

Land that isn’t put into crop production is harvested for hay, and sometimes protected if it harbors natural prairie vegetation.

Why is this land so flat, and where do those low elevation hills come from?

My ex-geologist husband explained that this area which is in the wide floodplain of the Red River (bordering North Dakota and northern Minnesota) was once lake bottom of the giant glacial lake Agassiz, and the ridges are remnants of its beach shorelines, where gravel accumulated.  Lake Winnipeg and Lake Manitoba in Canada and Lake of the Woods that borders the U.S. and Canada are what remains of this giant lake of the past Ice Age.

There is an interesting geology story here.  According to Wikipedia, glacial lake Agassiz once covered an area larger than all the Great Lakes combined running from central Manitoba into central Miinnesota.  At its greatest extent it may have covered as much as 170,000 square miles, approximately the size of the Black Sea today.  At the end of the last glaciation around 8,200 years ago, when melting ice in Hudson Bay allowed drainage of lake Agassiz, sea levels may have risen as much as 10 feet with the addition of fresh water on top of denser seawater. This hydrologic change may have altered ocean currents as well as world weather patterns, and is thought to be responsible for the widespread flooding of inland seas that has led to numerous flood myths of prehistoric cultures.  Some researchers link the melting event and subsequent sea level rise with the expansion of agriculture in Europe.