Read the poem HERE.
Nemerov's poem celebrates forms that recur in nature, particularly those that can be described by relatively simple mathematical forms or equations:
To lay the logarithmic spiral onThe same equation might accurately depict objects or patterns that are quite diverse, as much so as sea-shells, leaves, and the flight paths of fighter planes and bugs flying to a light. The narrator likens this recognition to noticing something that many people miss, like people driving past a hawk in a tree without noticing, or driving into the sun on an partly cloudy day and not seeing bright sun dogs right in front of them.
Sea-shell and leaf alike, and see it fit,
Let me try to be more explicit about such forms, in nature and in mathematics. First, to see some graphs of common mathematical equations, take a look Paul's Online Math Notes, which feature graphs of common equations, many of which describe forms found in nature:
Here's 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 equals the the result of plugging a given value of x into the equation.
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 taken out 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, which is precisely 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 determines how fast the curve rises. More precisely, B is the value of x that gives gives y equal to half of that maximum value. So if B is large, the curve rises slowly; if B is small, the first part of the curve is very steep.
The muscle oxygen carrier myoglobin is a small protein similar to one of the four chains of hemoglobin (see Biochemistry for Citizens, Unit 2A). 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 determines how fast the curve rises. More precisely, B is the value of x that gives gives y equal to half of that maximum value. So if B is large, the curve rises slowly; if B is small, the first part of the curve is very steep.
The muscle oxygen carrier myoglobin is a small protein similar to one of the four chains of hemoglobin (see Biochemistry for Citizens, Unit 2A). 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% oxymyoglobin), 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 in the waters of the cell 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
In Science and Human Values, Jacob Bronowski described creativity as the discovery of unity in variety, and appreciation as the recognition of that unity in a creative work. Do your think that we feel the presence of something creative when we suddenly see connections between things that previously seemed completely unrelated?
Another short poem comes to mind:
Poem: "Because You Asked about the Line Between Prose and Poetry", Howard Nemerov
Read the poem HERE.
In this poem (whose title is an essential part), Nemerov draws a likeness between two lines: the line between prose and poetry, and the line between rain and snow. I am keeping the narrator's ideas parallel, as he does: prose paired with rain, poetry with snow. This seems a logical pairing, the compact and orderly snow joined with poetry, which is often pithy and highly organized. Rain is more formless, and the form and organization of prose is often harder to describe or discern. Bronowski's view of creativity is certainly exemplified in this likeness drawn between forms of weather and forms of literature.
In that "moment that you could not tell", the distinction between rain and snow is blurred. Can you think of works of literature that blur the distinction between prose and poetry? Examples might be prose that has the grace and beauty of poetry, or blank verse that seems little different from prose except perhaps for the placement of lines and words on the page.
Finally, in "Figures of Thought, Nemerov's stinging second stanza raises the unsettling possibility that our own seemingly lofty and intricate behavior might boil down to some simple law, just beyond our grasp.
It may diminish some our dry delight
To wonder if everything we are and do
Lies subject to some little law like that;
Hidden in nature, but not deeply so.
••••••••••••
* A would-be poet should be willing to sell their soul just to create so good a line in a lifetime of writing.
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 in the waters of the cell 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
... same necessityNemerov goes on to say,
Ciphered in forms diverse and otherwise
Without kinship... *
– that is the beautifulHow do you interpret this statement? Is some of what we find beautiful, or simply arresting, in art or in nature, the result of recognizing familiar forms in unexpected contexts?
In Nature as in art, not obvious,
Not inaccessible, but just between.
In Science and Human Values, Jacob Bronowski described creativity as the discovery of unity in variety, and appreciation as the recognition of that unity in a creative work. Do your think that we feel the presence of something creative when we suddenly see connections between things that previously seemed completely unrelated?
Another short poem comes to mind:
Poem: "Because You Asked about the Line Between Prose and Poetry", Howard Nemerov
Read the poem HERE.
In this poem (whose title is an essential part), Nemerov draws a likeness between two lines: the line between prose and poetry, and the line between rain and snow. I am keeping the narrator's ideas parallel, as he does: prose paired with rain, poetry with snow. This seems a logical pairing, the compact and orderly snow joined with poetry, which is often pithy and highly organized. Rain is more formless, and the form and organization of prose is often harder to describe or discern. Bronowski's view of creativity is certainly exemplified in this likeness drawn between forms of weather and forms of literature.
In that "moment that you could not tell", the distinction between rain and snow is blurred. Can you think of works of literature that blur the distinction between prose and poetry? Examples might be prose that has the grace and beauty of poetry, or blank verse that seems little different from prose except perhaps for the placement of lines and words on the page.
Finally, in "Figures of Thought, Nemerov's stinging second stanza raises the unsettling possibility that our own seemingly lofty and intricate behavior might boil down to some simple law, just beyond our grasp.
It may diminish some our dry delight
To wonder if everything we are and do
Lies subject to some little law like that;
Hidden in nature, but not deeply so.
••••••••••••
* A would-be poet should be willing to sell their soul just to create so good a line in a lifetime of writing.
