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DN on lemniscates
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----- Original Message -----
From: DMITRI NABOKOV
To: D. Barton Johnson
Sent: Monday, October 06, 2003 7:45 AM
Reply to Carolyn, Mary, and Bekah, with thanks.
The equation of the Bernoullian lemniscate is (x²+y²)²=a²(x²-y²). That of the Boothian is (x²+y²)²=a²x² ± b²y² .
The latter is a generalization of the former. It is true that any such line has, in theory, no thickness, but is simply a conventional representation of an abstract mathematical expression. It is also true that a bicycle's tires, if they leave double traces on a sandy surface, would tend to do so on a curved, rather than relatively straight, section of the figure, if we are talking cycles propelled by human power. One could imagine, for the sake of argument, the tracks resulting from throttle, or simultaneous throttle and brake-induced oversteer or understeer , used, while still on the straight, to set up a dirt (motor) bike, dirt-track race car, or rally car for the turn ahead, but such considerations are irrelevant here. Nevertheless, I have a graphic representation of a Bernoullian lemniscate that distinctly thickens in the relatively straight NE-SW sector. It was conjecture on my part that Nabokov might have been somehow influenced by such a graphic depiction -- a typographical quirk, perhaps. Meanwhile, hats off to "Bekah"of the Pynchon List who has supplied an image of a perfectly good figure-eight being performed by a helmeted Cub Scout, apparently on his way to a Bear rating in Bicycle Safety. As a former Cub Scout myself, I was touched, but the figure did not appear to be a lemniscate. However, Bekah does attach several images and formulae for a variety of lemniscates, some of them undoubtedly feasible on an expertly ridden bike, and that, if we discount the thickening-line option, may include a genre of design VN may have had in mind when he used the word. The first one of the series, incidentally, while unlikely on the part of a recreational cyclist, is the kind of quadrifolium the skater performs in the promised poem. My lemning had better cease before I skate onto very thin ice.
Carolyn's ornate historical marvels are fascinating.
Best to all,
DN
----- Original Message -----
From: DMITRI NABOKOV
To: D. Barton Johnson
Sent: Monday, October 06, 2003 7:45 AM
Reply to Carolyn, Mary, and Bekah, with thanks.
The equation of the Bernoullian lemniscate is (x²+y²)²=a²(x²-y²). That of the Boothian is (x²+y²)²=a²x² ± b²y² .
The latter is a generalization of the former. It is true that any such line has, in theory, no thickness, but is simply a conventional representation of an abstract mathematical expression. It is also true that a bicycle's tires, if they leave double traces on a sandy surface, would tend to do so on a curved, rather than relatively straight, section of the figure, if we are talking cycles propelled by human power. One could imagine, for the sake of argument, the tracks resulting from throttle, or simultaneous throttle and brake-induced oversteer or understeer , used, while still on the straight, to set up a dirt (motor) bike, dirt-track race car, or rally car for the turn ahead, but such considerations are irrelevant here. Nevertheless, I have a graphic representation of a Bernoullian lemniscate that distinctly thickens in the relatively straight NE-SW sector. It was conjecture on my part that Nabokov might have been somehow influenced by such a graphic depiction -- a typographical quirk, perhaps. Meanwhile, hats off to "Bekah"of the Pynchon List who has supplied an image of a perfectly good figure-eight being performed by a helmeted Cub Scout, apparently on his way to a Bear rating in Bicycle Safety. As a former Cub Scout myself, I was touched, but the figure did not appear to be a lemniscate. However, Bekah does attach several images and formulae for a variety of lemniscates, some of them undoubtedly feasible on an expertly ridden bike, and that, if we discount the thickening-line option, may include a genre of design VN may have had in mind when he used the word. The first one of the series, incidentally, while unlikely on the part of a recreational cyclist, is the kind of quadrifolium the skater performs in the promised poem. My lemning had better cease before I skate onto very thin ice.
Carolyn's ornate historical marvels are fascinating.
Best to all,
DN