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Works discussed: "Figures of Thought" and "Because You Asked About the Line
Between Prose and Poetry", by Howard Nemerov; "From far, from eve and morning", by A.E. Housman.
For comments about the Housman poem, click on First Poem, in the right-hand column.
Here are more thoughts on the notion of mathematical forms found in nature.
••••••••••••
Passages from "Figures of Thought":
To lay the logarithmic spiral on
Sea-shell and leaf alike, and see it fit,
...
...
How secret that is, and how privileged
One feels to find the same necessity
Ciphered in forms diverse and otherwise
Without kinship –– that is the beautiful
In Nature as in art, not obvious,
Not inaccessible, but just between.
••••••••••••
Nature gives us many examples of form that can be described by relatively simple mathematical equations. You can find some examples on this web page:
This page of Paul's Online Math Notes features graphs of common equations, many of which are forms found in nature.
NOTE: If the following math gets too thick for you to plow, then just try to get the gist of it, but please think about the questions to the bottom, the paragraph that begins, "So this little equation... ."
NOTE: If the following math gets too thick for you to plow, then just try to get the gist of it, but please think about the questions to the bottom, the paragraph that begins, "So this little equation... ."
How to read Paul's examples: Beginning each section is an algebraic equation whose graph is then shown (don't worry about what's in between). If you are unfamiliar with graphing, here is how the graph is related to the equation. Using Example 1, pick a value of x (say, zero) from the horizontal axis, plug that value into the equation, and calculate y, you get y = 3. So you plot a point at (x,y) = (0,3), which will be at 3 on the vertical axis (where x = 0). If you do this for x = 1, then x = 2, and so forth, the points you plot will all lie on, or determine, the red line shown. In other words, the line is made of all points (x,y) for which y = –(2/5)x + 3. In like manner, the other curves comprise all points for which y = the function of x given in the equation for that example.
I believe that Descartes first conceived this bridge between algebra [such as y = –(2/5)x + 3] and geometry [in this case, a straight line]. But don't quote me on that, because I don't have a good head for history—especially the parts that happened in the past.
I believe that Descartes first conceived this bridge between algebra [such as y = –(2/5)x + 3] and geometry [in this case, a straight line]. But don't quote me on that, because I don't have a good head for history—especially the parts that happened in the past.
Where are some of these forms found in nature? For instance, Example 3, the parabola, accurately describes the flight of a ball or projectile launched upward and falling back down to earth. When I throw a tennis ball (left to right on the graph) for Darwin to chase, the ball follows the part of this path that begins with my launch angle. If I throw the ball at a low angle, its path is a section of the top of this curve; if I throw it high to give Darwin a chance to run under it and hear where it lands, its path is more like this entire curve (and Darwin is more likely to find it and bring it back).
The "logarithmic spiral" that Nemerov uses as his example, is something like Example 9 or 10. Other common forms include the ellipse, Example 6, the path that planets follow around the sun, and the circle, which is a special case of the ellipse, and which occurs in the ripples of water that move outward from a rock dropped in a pool.
Not shown on Paul's nice page is surprisingly simple equation, the square hyperbola, that describes diverse situations, including the amount of binding of binding of oxygen to myoglobin in muscle; the rate of reaction of bacterial cell walls with the defensive enzyme lysozyme, found in tears and mucus; and the relationship between the numbers of prey animals and the number of predators that are adequately fed by them.
The equation is
y = A [x/(B + x),
In this equation, A represents the maximum value of that y can take, and B represents the value of that determines how fast the curve rises.
The muscle oxygen carrier myoglobin is a small protein similar to one of the four chains of hemoglobin). When oxygen is present, it binds noncovalently (weak bonds) to myoglobin, giving oxy-myoglobin. The percentage of myoglobin converted to oxymyoglobin depends on how much oxygen is present, as shown here:
Not shown on Paul's nice page is surprisingly simple equation, the square hyperbola, that describes diverse situations, including the amount of binding of binding of oxygen to myoglobin in muscle; the rate of reaction of bacterial cell walls with the defensive enzyme lysozyme, found in tears and mucus; and the relationship between the numbers of prey animals and the number of predators that are adequately fed by them.
The equation is
y = A [x/(B + x),
In this equation, A represents the maximum value of that y can take, and B represents the value of that determines how fast the curve rises.
The muscle oxygen carrier myoglobin is a small protein similar to one of the four chains of hemoglobin). When oxygen is present, it binds noncovalently (weak bonds) to myoglobin, giving oxy-myoglobin. The percentage of myoglobin converted to oxymyoglobin depends on how much oxygen is present, as shown here:
Vertical Axis (y): Percent of Oxy-myoglobin
Horizontal Axis (x): Amount of Oxygen Present
The equation of this curve is y = 100[x/(5+x)]. The number 100 is the maximum value of y (you can't have more than 100% myoglobin), and 5 is the amount of oxygen that produces 50% oxymyoglobin. So the number 5 tells us how fast this curve rises as the amount of oxygen increases.
You can see from this curve that, when oxygen is plentiful, most of the myoglobin is carrying oxygen, but when oxygen levels are low, most of the myoglobin is oxygen-free, which means that oxygen is available to the muscle. In other words, as muscle consumes oxygen and reduces its level, myoglobin gives it up (this is called equilibrium or reversible binding). It turns out that myoglobin makes oxygen much more soluble than free oxygen, so it increases the amount of oxygen that can dissolve, and thus be readily available, in the cytoplasm of muscle cells.
Probably more than you wanted to know, but an example of how the action of a carrier protein reacts to changing availability of its cargo. The same curve applies to the binding of antibodies to antigens (foreign substances), and to many other situations.
So the equation of the square hyperbola is one example of Nemerov's
... the same necessity
Ciphered in forms diverse and otherwise
Without kinship ...
Nemerov goes on to say that this is "the beautiful in Nature as in art". How do you interpret this statement? Is some of what we find beautiful, or simply arresting, in art the result of recognizing familiar forms in unexpected contexts? Do we feel the presence of something creative when we suddenly see connections between things that previously seemed completely unrelated?
You can see from this curve that, when oxygen is plentiful, most of the myoglobin is carrying oxygen, but when oxygen levels are low, most of the myoglobin is oxygen-free, which means that oxygen is available to the muscle. In other words, as muscle consumes oxygen and reduces its level, myoglobin gives it up (this is called equilibrium or reversible binding). It turns out that myoglobin makes oxygen much more soluble than free oxygen, so it increases the amount of oxygen that can dissolve, and thus be readily available, in the cytoplasm of muscle cells.
Probably more than you wanted to know, but an example of how the action of a carrier protein reacts to changing availability of its cargo. The same curve applies to the binding of antibodies to antigens (foreign substances), and to many other situations.
So the equation of the square hyperbola is one example of Nemerov's
... the same necessity
Ciphered in forms diverse and otherwise
Without kinship ...
Nemerov goes on to say that this is "the beautiful in Nature as in art". How do you interpret this statement? Is some of what we find beautiful, or simply arresting, in art the result of recognizing familiar forms in unexpected contexts? Do we feel the presence of something creative when we suddenly see connections between things that previously seemed completely unrelated?
