Skid Patch Theorem
11-02-06 | 03:08 PM
#1
Thread Starter
Junior Member
Joined: Jan 2006
Posts: 17
Likes: 0
From: Madison, WI
Bikes: 2008 ZR Cycles track, 1985 schwinn peloton, 1983 univega gran turismo, fixed gear beater
Skid Patch Theorem
Theorem:
(2) is generally not known and disproves several conjectures seen recently.
Proof of (1):
Let a / b be the reduced gear ratio (that is, a and b are integers with no common divisors other than 1). Then,
(1) With single-sided skidding, there are b skid patches, and
(2) Ambidexterous skidding doubles the number of skid patches if and only if a is odd.
(1) is already well-known. It is proved here for completeness.(1) With single-sided skidding, there are b skid patches, and
(2) Ambidexterous skidding doubles the number of skid patches if and only if a is odd.
(2) is generally not known and disproves several conjectures seen recently.
Proof of (1):
Turning the pedals through one revolution turns the wheel through a / b revolutions. Turning the pedals through b revolutions turns the wheel through b * (a / b) = a revolutions. That is, after b pedal revolutions, the wheel is returned to the same position it was originally (since a is an integer). So there must be no more than b skid patches, since the same cycle of b wheel positions will be repeated through every b pedal revolutions.
Now suppose that two of the intermediate wheel positions were the same, say, after i and j pedal revolutions (0 <= i < j < b). Then j - i pedal revolultions also returns the wheel to its original position, so (j - i) * a / b is an integer. Thus b must evenly divide (j - i) * a / b. However, a and b have no common divisors, so b must evenly divide j - i. But j - i is less than b, so this cannot happen. Therefore, all b of the intermediate wheel positions (after 0, 1, 2, ..., and b-1 pedal revolutions) are different. So there must be no fewer than b skid patches.
There are no more than b skid patches and there are no fewer than b skid patches, so there must be exactly b skid patches.
Proof of (2):Now suppose that two of the intermediate wheel positions were the same, say, after i and j pedal revolutions (0 <= i < j < b). Then j - i pedal revolultions also returns the wheel to its original position, so (j - i) * a / b is an integer. Thus b must evenly divide (j - i) * a / b. However, a and b have no common divisors, so b must evenly divide j - i. But j - i is less than b, so this cannot happen. Therefore, all b of the intermediate wheel positions (after 0, 1, 2, ..., and b-1 pedal revolutions) are different. So there must be no fewer than b skid patches.
There are no more than b skid patches and there are no fewer than b skid patches, so there must be exactly b skid patches.
As above, turning the pedals through one revolution turns the rear wheel through a / b revolutions. Turning the pedals through one half-revolution turns the rear wheel through half as many revolutions. So the number of skid patches with ambidexterous skidding should be the same as that with single-sided skidding on a gear ratio half as large. Now to apply (1) to this situation, we need to know how 1/2 * a/b reduces as an integer ratio. This depends on whether a is even or odd. If a is even, (a/2) / b is the reduced ratio, so there are b skid patches, as in the single-sided case. If a is odd, a / (2b) is the reduced ratio, so there are 2b skid patches.
)
11-02-06 | 05:09 PM
#5
This new learning amazes me, Sir Bedevere. Explain again how sheep's bladders may be employed to prevent earthquakes.
__________________
This is Africa, 1943. War spits out its violence overhead and the sandy graveyard swallows it up. Her name is King Nine, B-25, medium bomber, Twelfth Air Force. On a hot, still morning she took off from Tunisia to bomb the southern tip of Italy. An errant piece of flak tore a hole in a wing tank and, like a wounded bird, this is where she landed, not to return on this day, or any other day.
This is Africa, 1943. War spits out its violence overhead and the sandy graveyard swallows it up. Her name is King Nine, B-25, medium bomber, Twelfth Air Force. On a hot, still morning she took off from Tunisia to bomb the southern tip of Italy. An errant piece of flak tore a hole in a wing tank and, like a wounded bird, this is where she landed, not to return on this day, or any other day.
11-02-06 | 08:57 PM
#7
blacksheep the blemish
Joined: Nov 2004
Posts: 1,063
Likes: 0
From: Portland/Greendale
Bikes: 1973 Schwinn World Voyageur (manufactured by panasonic), Italvega Super Speciale (fixed, primary ride now), Kona 2004 JTS 10 spd
Jesus, I was trying to blow off my number theory homework not do extra curricular work.
Anyways, nice proof.
Anyways, nice proof.
11-03-06 | 08:31 AM
#12
Arizona Dessert
Joined: Jun 2004
Posts: 15,029
Likes: 2,170
From: AZ
Bikes: Cannondale SuperSix, Lemond Poprad. Retired: Jamis Sputnik, Centurion LeMans Fixed, Diamond Back ascent ex
Maybe its time to update my SP chart and put the ones to double for ambi in italics or somesuch. Last update was may-05
03-27-07 | 12:17 PM
#14
Senior Member
Joined: Feb 2007
Posts: 228
Likes: 0
From: Madison, WI
I've added a "skid patch analysis" function to rabbit showing the conclusion of Fraction's SPT is valid.
Check it out at https://software.bareknucklebrigade.com if interested.
Check it out at https://software.bareknucklebrigade.com if interested.
03-27-07 | 07:33 PM
#17
Senior Member
Joined: Jan 2005
Posts: 285
Likes: 0
theres a book called https://en.wikipedia.org/wiki/Cryptonomicon it has a cool discription of how code machines work n it uses a bike to describe it.
03-27-07 | 09:26 PM
#18
Senior Member
Joined: Jun 2003
Posts: 1,155
Likes: 0
From: Atlanta
Bikes: Zeus (Razesa) tarck, Giant TCR road, Eddy Merckx road, Fuji Touring Series IV for everything else
Originally Posted by thatcher
theres a book called https://en.wikipedia.org/wiki/Cryptonomicon it has a cool discription of how code machines work n it uses a bike to describe it.
This was when I was a math major but yet to get into bikes.
03-28-07 | 01:39 AM
#19
Sheldon Brown's posse
Joined: Jul 2006
Posts: 1,046
Likes: 0
From: Oz-tray-lee-ah
Bikes: BMC SL01, XtC, Rhythm GX and a frankenstein avalanche 2.0
ok. I am having a little trouble understanding exactly what a skid patch is. I'll take a guess and say every rotation of the pedals you lock up the wheels once or twice (single-sided or ambidextrous)?
03-28-07 | 05:49 AM
#20
antisocialite
Joined: Apr 2006
Posts: 3,385
Likes: 1
ask and ye shall be linked:
https://sheldonbrown.com/fixed.html#skid
edit: if the theorem above is correct, and it looks like it is, the sentence should read:
If you are an ambidextrous skidder, and the simplified ratio has an even numerator, your number of skid patches will be the same.
If you are an ambidextrous skidder, and the numerator and denominator is odd, the number of possible skid patches will be doubled.
to be fair, i was the one who called sheldon out on this originally, and being the reasonable person he is, he changed his site to reflect it. turns out we were both wrong
https://sheldonbrown.com/fixed.html#skid
edit: if the theorem above is correct, and it looks like it is, the sentence should read:
If you are an ambidextrous skidder, and the simplified ratio has an even numerator, your number of skid patches will be the same.
If you are an ambidextrous skidder, and the numerator and denominator is odd, the number of possible skid patches will be doubled.
to be fair, i was the one who called sheldon out on this originally, and being the reasonable person he is, he changed his site to reflect it. turns out we were both wrong
03-28-07 | 05:51 AM
#21
jack of one or two trades
Joined: Jun 2005
Posts: 5,640
Likes: 0
From: Suburbia, CT
Bikes: Old-ass gearie hardtail MTB, fix-converted Centurion LeMans commuter, SS hardtail monster MTB
Originally Posted by shogun17
ok. I am having a little trouble understanding exactly what a skid patch is.
03-28-07 | 06:45 AM
#23
oldsprinter
Joined: Nov 2006
Posts: 935
Likes: 0
From: Tokyo
Originally Posted by dirtyphotons
ask and ye shall be linked:
https://sheldonbrown.com/fixed.html#skid
edit: if the theorem above is correct, and it looks like it is, the sentence should read:
If you are an ambidextrous skidder, and the simplified ratio has an even numerator, your number of skid patches will be the same.
If you are an ambidextrous skidder, and the numerator and denominator is odd, the number of possible skid patches will be doubled.
to be fair, i was the one who called sheldon out on this originally, and being the reasonable person he is, he changed his site to reflect it. turns out we were both wrong
https://sheldonbrown.com/fixed.html#skid
edit: if the theorem above is correct, and it looks like it is, the sentence should read:
If you are an ambidextrous skidder, and the simplified ratio has an even numerator, your number of skid patches will be the same.
If you are an ambidextrous skidder, and the numerator and denominator is odd, the number of possible skid patches will be doubled.
to be fair, i was the one who called sheldon out on this originally, and being the reasonable person he is, he changed his site to reflect it. turns out we were both wrong
03-28-07 | 10:20 AM
#25
Senior Member
Joined: Sep 2006
Posts: 3,744
Likes: 1
From: Van BC
Originally Posted by dirtyphotons
ask and ye shall be linked:
https://sheldonbrown.com/fixed.html#skid
edit: if the theorem above is correct, and it looks like it is, the sentence should read:
If you are an ambidextrous skidder, and the simplified ratio has an even numerator, your number of skid patches will be the same.
If you are an ambidextrous skidder, and the numerator and denominator is odd, the number of possible skid patches will be doubled.
to be fair, i was the one who called sheldon out on this originally, and being the reasonable person he is, he changed his site to reflect it. turns out we were both wrong
https://sheldonbrown.com/fixed.html#skid
edit: if the theorem above is correct, and it looks like it is, the sentence should read:
If you are an ambidextrous skidder, and the simplified ratio has an even numerator, your number of skid patches will be the same.
If you are an ambidextrous skidder, and the numerator and denominator is odd, the number of possible skid patches will be doubled.
to be fair, i was the one who called sheldon out on this originally, and being the reasonable person he is, he changed his site to reflect it. turns out we were both wrong
Originally Posted by fraction
Let a / b be the reduced gear ratio (that is, a and b are integers with no common divisors other than 1). Then,
...
(2) Ambidexterous skidding doubles the number of skid patches if and only if a is odd.
...
(2) Ambidexterous skidding doubles the number of skid patches if and only if a is odd.