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Fw: Bernoulli lemniscate (Pale Fire)
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----- Original Message -----
From: Mary Krimmel
Message requiring your approval (49 lines) ------------------ Dmitri Nabokov wrote:
"Lemniscate" has various meanings and various relatives. The two definitions that interest us are 1) the Bernoullian lemniscate and 2) the Boothian lemniscate which is a generalization of the Bernoullian. I have the precise formulae for both but shall not overload this posting with math. Suffice it to say that the Bernoullian variety is neither simply an eight snoozing on its side nor the infinity symbol. If one examines it carefully one will notice that part of the central section (roughly from NE to SW) is thicker than the rest of the figure. This may be interpreted as the zone where the track was thickened by the imperfect overlap of two bicycle tires, which just might have been what Nabokov intended to convey by his choice of the word. The skater's joy Phil mentions is exemplified by a poem of my father's -- which I have translated for my collection and shall post as soon as it has come out in Aretè -- in which a deft blade can even design a flower. So maybe we should set lemnicycling aside and instead lemniskate.
A couple of questions/comments:
What is a Boothian lemniscate? I am aware of generalizations of the Bernoulli lemniscate, but do not know which is called Boothian. Why does it interest us? (It interests me just because it's mentioned and I don't know what it is.)
Bicycle tire tracks might well be thicker at one place than another, but wouldn't one expect them to be thicker at the more curved outsides of our lemniscate than at the less curved central section? (Not of a figure eight, though, where the absolute curve is constant.) The Bernoulli lemniscate is a line and has no thickness at all, in any part. What does one exam carefully? Perhaps the thickening is observed in the not-quite-perpendicular shadow of a double loop of ribbon, the original lemniscus, which is thicker in the third dimension where it crosses itself.
The suggestion that "we should set lemnicycling aside and lemniskate" is more than worthy of any Nabokov. Thanks for that touch.
(Phil had written: "...I have ridden a bicycle and just remember the joyous going nowhere of riding round in circles (more or less) that followed my first acquisition of balance on a machine too big for my childish frame. Going round in circles is apt to make one dizzy so progress to rough figures of eight prolongs the going nowhere joy. I'm sure this is the same joy that skaters feel.")
I look forward to reading the translation of the poem exemplifying that joy. Thank you for offering to post it here.
Mary Krimmel