## Math: Fleeting Thoughts

For the discussion of math. Duh.

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Eebster the Great
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### Re: Math: Fleeting Thoughts

I was thinking of the MLB, where even slow curves typically exceed 30 m/s and cut fastballs can get close to 40.

According to Alam et al., flow starts to become turbulent around 40 km/h and becomes fully turbulent by 120 km/h, or about 75 mph. In the MLB, the average curveball is travelling at 76.4 mph after leaving the pitcher's hand and reaches a minimum speed of 70.4 mph before reaching the catcher's glove, easily within the turbulent regime. Other breaking balls like sliders of course travel faster, and fastballs average 90.9 mph off the glove and 83.2 at the plate (which is actually slower than I would have expected).

Carlington
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### Re: Math: Fleeting Thoughts

I've been playing Euclidea lately, which has been doing a fine job of how much compass and straight-edge construction I've forgotten since geometry. It's been good fun to muddle my way through, and I plan to get some of the dev's other apps and to improve my solutions once I've finished.
I've reached an impasse, though, which I haven't been able to conquer after a weeks' worth of trying. I've been given a circle, with centre marked, and a point outside the circle. I need to construct a secant line through that point, such that the circle bisects the secant line, i.e the distance from the point to the first intersection with the circle should equal the distance from the first to the second intersection with the circle.

As it's a game and I'm enjoying it, I still want to get the warm fuzzy dopamine hit from the reward centre of my brain, so I don't want the solution out and out spoiled - that said, it would save me some sleep if I could be prodded with something that points in the direction of the solution.
My main serious attempts (not counting just drawing lines and circles and connecting their intersections and hoping) have been:
- construct the diameter of the circle through the point, and the tangent of the circle through the point, then bisect the angle so formed
- as above, but instead connect the centre to the tangent point, then bisect that line segment and connect the midpoint to the point given
- construct the midpoint of the point and the circle's centre, and then construct a tangent from that midpoint (this was impossible, as the midpoint fell within the circle)
- construct the diameter, and any other secant line through the point. Construct a line through the centre (midpoint of the diameter) and the midpoint of the other secant line. Continue this line until it intersects with the circle, and then construct the secant from the point through this point of intersection.
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Eebster the Great: What specifically is moving faster than light in these examples?
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phlip
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### Re: Math: Fleeting Thoughts

I'm not sure how spoilery this suggestion is compared to what you want, so I've broken it up into two...
Spoiler:
When I'm working on this sort of thing, I'll usually work backwards from the solution, rather than forward from the question... asking not "what can I construct from what I have?" but "what would be useful to construct what I need?"

So I'll start with the diagram with the solution already drawn in, and add lines/etc built off that until I've found something that I can construct from the initial givens, then adapt that whole thing in reverse into an actual construction.

(The next spoiler has what I would try as the first step of this reverse-construction. Now what is necessarily the right choice, but just what I'd try.)
Spoiler:
In particular in this case, the whole bisected-inside-and-outside thing makes me want to mirror everything so that the two halves of the secant line match. Have a perpendicular bisector of the secant line, which we use as an axis of symmetry. A reflection of the circle on that axis.

So that if P is our original point, and the secant cuts the original circle at A and then ends at B (so that PA = AB)... we draw our perpendicular at A, have B as a reflection of P, and have a circle that goes through P and A, as a reflection of our original circle that goes through A and B.

Of course, we still don't know how to figure out where that circle or axis need to be, but that's where I'd start.

Code: Select all

`enum ಠ_ಠ {°□°╰=1, °Д°╰, ಠ益ಠ╰};void ┻━┻︵​╰(ಠ_ಠ ⚠) {exit((int)⚠);}`
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bentheimmigrant
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### Re: Math: Fleeting Thoughts

So, I've essentially got a simplified version of the sofa problem... I just want to get a piece of wood through into a gap behind a fake wall. What is the longest piece I can get through, assuming that it is touching the floor, top of the opening, and the real wall at the back simultaneously at the tightest point?

The real wall is 0.9m behind the fake wall, and the opening is 0.3m tall.

I tried coming up with an equation for length wrt the angle of the wood against the floor, and came up with L = (0.9 - (0.3/tanx))/cosx

Which I suspect is wrong, but I'm not sure how...

Anyways, if memory serves, I should find dL/dx, which should be 0 at the point where the length is a minimum. But this is hard, and Wolfram Alpha gave me a fairly complicated solution, and I couldn't get anything to work.

But all the while this seems much simpler than I've made it... Halp?

Edit: So apparently (and not surprisingly), this is a specific problem other people have addressed. Amazing what a good night's sleep and some fresh googling can do.
https://ckrao.wordpress.com/2010/11/07/ ... r-problem/

Would still be interesting to see if anyone can resolve the trigonometric approach.
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liberonscien
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### Re: Math: Fleeting Thoughts

I think the term for taking something to the fourth power should be "hypercube".

Thesh
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### Re: Math: Fleeting Thoughts

Shouldn't that be "tesseract"?
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Eebster the Great
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### Re: Math: Fleeting Thoughts

I remember some sci fi short story using the term "quartic femtometer" in casual conversation as an exaggeration to refer to an extremely tiny region of spacetime. Personally, I thought "quartic femtosecond" (or better yet, yoctosecond) would be superior in that it is much smaller, but I'm sure the author felt this would be understood by almost nobody.

Thesh wrote:Shouldn't that be "tesseract"?

Or 4-cube? Or 4-regular-orthotope? Doesn't really have the same ring to it.

Carlington
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### Re: Math: Fleeting Thoughts

Hypercube should be pronounced with the same stress pattern as hyperbola.
Kewangji: Posdy zwei tosdy osdy oady. Bork bork bork, hoppity syphilis bork.

Eebster the Great: What specifically is moving faster than light in these examples?
doogly: Hands waving furiously.

Please use he/him/his pronouns when referring to me.

Zohar
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### Re: Math: Fleeting Thoughts

high-PER-queue-BEH?
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Carlington
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### Re: Math: Fleeting Thoughts

Yes, exactly that.
Kewangji: Posdy zwei tosdy osdy oady. Bork bork bork, hoppity syphilis bork.

Eebster the Great: What specifically is moving faster than light in these examples?
doogly: Hands waving furiously.

Please use he/him/his pronouns when referring to me.

WibblyWobbly
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### Re: Math: Fleeting Thoughts

Eebster the Great wrote:I remember some sci fi short story using the term "quartic femtometer" in casual conversation as an exaggeration to refer to an extremely tiny region of spacetime. Personally, I thought "quartic femtosecond" (or better yet, yoctosecond) would be superior in that it is much smaller, but I'm sure the author felt this would be understood by almost nobody.

Thesh wrote:Shouldn't that be "tesseract"?

Or 4-cube? Or 4-regular-orthotope? Doesn't really have the same ring to it.

I like how Wikipedia's entry on n-orthotopes begins with "In geometry, an n-orthotope (also called a hyperrectangle or a box) ...

Can't we at least call it a hyperbox? An n-box?

Copper Bezel
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### Re: Math: Fleeting Thoughts

That doesn't tell you that it has equal sides, though. Which could make for some strange exponent behavior and strange units of measure.
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Eebster the Great
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### Re: Math: Fleeting Thoughts

Copper Bezel wrote:That doesn't tell you that it has equal sides, though. Which could make for some strange exponent behavior and strange units of measure.

That's why I said "regular". But yeah, the simplest term would be "4-cubed," which is a rather silly way of doing things.

Carlington
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### Re: Math: Fleeting Thoughts

I was watching this Numberphile video about Pascal's triangle, and learned yet more things about it. In particular, I really liked the section starting here. When she hadn't even started drawing the lines in yet I was starting to notice the pattern and was genuinely saying to my computer screen "If this is Sierpinski's Triangle, I swear to god..." and then it was! Is there anything this triangle can't do?
Kewangji: Posdy zwei tosdy osdy oady. Bork bork bork, hoppity syphilis bork.

Eebster the Great: What specifically is moving faster than light in these examples?
doogly: Hands waving furiously.

Please use he/him/his pronouns when referring to me.

cyanyoshi
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### Re: Math: Fleeting Thoughts

Carlington wrote:I was watching this Numberphile video about Pascal's triangle, and learned yet more things about it. In particular, I really liked the section starting here. When she hadn't even started drawing the lines in yet I was starting to notice the pattern and was genuinely saying to my computer screen "If this is Sierpinski's Triangle, I swear to god..." and then it was! Is there anything this triangle can't do?

Ah yes, good old Rule 60.

Xenomortis
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### Re: Math: Fleeting Thoughts

Carlington wrote:I was watching this Numberphile video about Pascal's triangle, and learned yet more things about it. In particular, I really liked the section starting here. When she hadn't even started drawing the lines in yet I was starting to notice the pattern and was genuinely saying to my computer screen "If this is Sierpinski's Triangle, I swear to god..." and then it was! Is there anything this triangle can't do?

You get similar patterns when considering modulus of any prime (Sierpinski's triangle with n(n+1)/2 duplicates instead of 2, for prime n).
Actually, it works for any number, not just primes, but it's a little more complicated then.

Qaanol
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### Re: Math: Fleeting Thoughts

One degree is approximately 1.75 percent
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cyanyoshi
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### Re: Math: Fleeting Thoughts

That reminds me of something. I was fiddling around with some geometry-based algorithm that works well when an angle is a rational multiple of 2*pi. As a test, I saw what would happen if the angle was 1 radian. Surprisingly, everything worked out as if the angle was (7/44)*2*pi instead, but then it hit me that 22/7 is a very well-known approximation for pi. I felt slightly dumb afterwards.

gd1
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### Re: Math: Fleeting Thoughts

Heh, peach vise functions.

piecewise :p
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gd1
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### Re: Math: Fleeting Thoughts

With regards to which part of mathematics I enjoy most, I'm somewhat partial to fractions.
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Xanthir
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### Re: Math: Fleeting Thoughts

Qaanol wrote:
One degree is approximately 1.75 percent

In what way? It's not 1.75 percent of a circle, or even of a quarter-arc.

(Vaguely related, 1px in CSS is about 1.25 arcminutes - the CSS length units are technically angle units, since they scale by viewing distance to subtend the same fraction of your view. ^_^)

(I know this is responding to a many-month-old post, but Qaanol has to answer for themself, dammit!)
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doogly
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### Re: Math: Fleeting Thoughts

Of a radian. Not to steal Qaanol's thunder though.
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Qaanol
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### Re: Math: Fleeting Thoughts

Xanthir wrote:
Qaanol wrote:
One degree is approximately 1.75 percent

In what way? It's not 1.75 percent of a circle, or even of a quarter-arc.

(Vaguely related, 1px in CSS is about 1.25 arcminutes - the CSS length units are technically angle units, since they scale by viewing distance to subtend the same fraction of your view. ^_^)

(I know this is responding to a many-month-old post, but Qaanol has to answer for themself, dammit!)

Doogly got it.

Being excessively pedantic, a radian is the ratio of arc length to radius length which is identically equal to 1, with no units. So a degree is just… the number π/180.
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### Re: Math: Fleeting Thoughts

I need to get my suroctonions tetrated soon.
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Thesh
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### Re: Math: Fleeting Thoughts

I think the round function is pretty poorly named:

Without rounding.

With rounding.
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Xanthir
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### Re: Math: Fleeting Thoughts

Qaanol wrote:
Xanthir wrote:
Qaanol wrote:
One degree is approximately 1.75 percent

In what way? It's not 1.75 percent of a circle, or even of a quarter-arc.

(Vaguely related, 1px in CSS is about 1.25 arcminutes - the CSS length units are technically angle units, since they scale by viewing distance to subtend the same fraction of your view. ^_^)

(I know this is responding to a many-month-old post, but Qaanol has to answer for themself, dammit!)

Doogly got it.

Being excessively pedantic, a radian is the ratio of arc length to radius length which is identically equal to 1, with no units. So a degree is just… the number π/180.

This is good pedantry, but Qaanol's wasn't. You can't just take any number and say it's a percentage by multiplying it by 100; percentages *mean* something, dang it!

(In this case, the correct answer is "a degree is approximately 1.75% of a radian". The fact that a radian is just the unitless value 1 doesn't mean you can just ignore it when describing what a degree is a % of.)

Thesh wrote:
I think the round function is pretty poorly named:

Without rounding.

With rounding.

Looks rounded to me, I don't see the problem.
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### Re: Math: Fleeting Thoughts

Qaanol
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### Re: Math: Fleeting Thoughts

Xanthir wrote:This is good pedantry, but Qaanol's wasn't. You can't just take any number and say it's a percentage by multiplying it by 100; percentages *mean* something, dang it!

(In this case, the correct answer is "a degree is approximately 1.75% of a radian". The fact that a radian is just the unitless value 1 doesn't mean you can just ignore it when describing what a degree is a % of.)

“1 percent” is the unitless number 1/100
“1 degree” is the unitless number π/180

I stand by my pedantry.
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ucim
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### Re: Math: Fleeting Thoughts

Qaanol wrote:“1 percent” is the unitless number 1/100
“1 degree” is the unitless number π/180

“1 percent” literally means "one per hundred", which is a ratio, which is a (rational) number, and therefore is the unitless number 1/100.

“1 degree” literally means "one (specific) part of a(n understood) whole, that whole being (in math) the circumference of a unit circle in the plane, and the specific part being 1/360 of it. And while the circumference's length equals 2π, the circumference is not itself identical to that number. It is a circumference, not a number.

Therefore it is not the unitless number π/180, but rather, the unitted value 1/360 of a unit circle's circumference (and all that comes from that).

There is no case (at least known to me) in which "one percent" does not mean one one-hundreth. Sometimes the thing it's one one-hundredth of is understood, but it is separate from the percent (concept) itself.

However, there are many cases where "one degree" does not mean, or come from, the number π/180. Degrees of the (musical) scale come to mind, as do degrees Rankine and degrees Celsius. In both cases the meaning of "one degree" derives from the idea of a specific part of a (specified or understood) whole. Even the PhD degree ultimately derives from this, and it most certainly is not a unitless number equal to π/180 (or anything else).

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Eebster the Great
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### Re: Math: Fleeting Thoughts

I think it is well accepted that 1% = 0.01. The question is the meaning of angle measure. The terse Complex Analysis text I just conveniently lost defined the radian measure according to the Taylor series that could most conveniently fit the desired properties. Clearly one measure is way more convenient than all others. But defining 1 rad = 1 is not a meaningless step and comes down to the chosen definition of the trig functions.

The geometric and calculus explanations of why radians are the best measure is very convincing, but if we insisted, we could still use any other measure if we really had to. The mathematical conclusions would not change. It would just be harder to explain weird angles to students.

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### Re: Math: Fleeting Thoughts

1 degree is about -25.56 dB. The units check out.

Flumble
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### Re: Math: Fleeting Thoughts

The degree is a unit of work, equal to about 3 years.

DavidSh
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### Re: Math: Fleeting Thoughts

I think it works better as "man-years" rather than just "years". (This is the gender-free meaning of "man", just meaning a worker.)

Thesh
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### Re: Math: Fleeting Thoughts

The gender-neutral term is "person-years".
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ucim
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### Re: Math: Fleeting Thoughts

No. It's "persibling-years".

Jose
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gmalivuk
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### Re: Math: Fleeting Thoughts

One's son shouldn't be one's sibling...
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Flumble
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### Re: Math: Fleeting Thoughts

In any case, let's not discount any non-human entities that may work for degrees. They're worker-years.

For some mathy thoughts: if you have a linear programming problem in k dimensions/properties and n variables, does your solution contain at most min{k,n} non-zero entries?
In particular, I have a large set of foodstuffs (vectors/variables) with a couple of nutrients (dimensions) in them, a minimum for each nutrient and I'm optimizing for price. I'd like to know beforehand if throwing LP at it will give a concise solution or tiny bits of everything. (My gut says it will, because for 3 foods and 2 nutrients, food C is a linear combination of A+B that is either cheaper or more expensive, so for any point in the feasible region, you can optimize by either removing A+B for more C until A/B runs out or adding more A+B until C runs out.)

Xenomortis
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### Re: Math: Fleeting Thoughts

gmalivuk wrote:One's son shouldn't be one's sibling...

I've got some friends that may argue against that.
Spoiler:
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doogly
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### Re: Math: Fleeting Thoughts

gmalivuk wrote:One's son shouldn't be one's sibling...

The moral injunctions of Exodus can really go take a hike. 2019, brah.
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### Re: Math: Fleeting Thoughts

Flumble wrote:In any case, let's not discount any non-human entities that may work for degrees. They're worker-years.

For some mathy thoughts: if you have a linear programming problem in k dimensions/properties and n variables, does your solution contain at most min{k,n} non-zero entries?
In particular, I have a large set of foodstuffs (vectors/variables) with a couple of nutrients (dimensions) in them, a minimum for each nutrient and I'm optimizing for price. I'd like to know beforehand if throwing LP at it will give a concise solution or tiny bits of everything.

Yes, that is correct, with minor caveats.
For a problem like "Minimize c'x subject to Ax >= b, x >=0", such as your diet problem (sometimes called the Prisoner's Diet Problem), if the cost vector c is positive there will be an optimal solution where at least n of the inequality constraints are tight. This represents a corner of the feasible region. If this solution has m non-zeros, then only n-m of the non-negativity constraints are tight. So at least m of the nutrition constraints Ax >= b must be tight. This is only possible if m is at most k.

Caveats are:
(1) If the costs c have some negative values, an optimal solution might involve infinite amounts of some foodstuffs.
(2) If you have multiple optimal solutions, some might have more non-zeros. For example, if you have two foodstuffs with quantities x and y, and only 1 nutrient constraint x+y >= 1, and the foodstuffs also have equal costs, x=1,y=0 is an optimum, as is x=0,y=1, as is the entire line x=alpha,y=1-alpha for 0 <= alpha <= 1. So, while some optimal solutions have min(k,n)=1 non-zero, other optimal solutions have 2 non-zeros.