## 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.

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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".
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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.

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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

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.

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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