Thales’ circles

“All is water”

Posts tagged math

18,332 notes

(We’re taking a calculus final. The TA is a well-known Lord of the Rings fan, and we’ve had running LotR jokes all semester.)
TA:
“Okay, guys, everyone look at me. We’ve been over the rules, but just in case: no notes, pencil your answers in on the scantron sheet, and graphing calculators only – no more ‘can I just used my cell phone’ nonsense.”
Student:
“[TA's name], my calculator batteries just died! What should I do?”
TA:
“Here, I’ve got a big box of spares.”
Student:
*struggling* “I can’t get this packaging open…”
Student 2:
“Here, I’ve got a pocket knife.”
TA:
“And I’ve got a pair of scissors if you need them.”
Student 3:
*from the back of the room* “OR MY AXE!”
(Everyone starts laughing.)
TA:
“The only axes allowed on the exam are in the graph section.”
(Everyone groans.)
TA:
“Oh, come on, you’re in a math class. Deal with the math jokes.”
(The professor enters with a stack of exams. With him are two exam proctors.)
Professor:
“Tolkien jokes already, [TA's name]?”
TA:
“Hey, I didn’t start it.”
(The professor starts handing stacks of exams to the TA and proctors.)
Professor:
“But I’m about to finish it. [TA], take these exams down the left flank. [Proctor 1], follow the desks down the center. [Proctor 2], take your exams right, along the wall.”
(At this point, many of the students have realized where this is going:
Theoden’s lines from ‘Return of the King.’)
Professor:
“Forth, and fear no problems! Solve! Solve, students of calculus! Points shall be taken, scores shall be splintered! A pencil day! A red-ink day! Until three thirty!”
(The professor pulls out a pencil, holding it out like a sword, and runs down the first row holding it out. Students hold up their pencils, hitting his as he passes.)
Professor:
“Solve now! Solve now! Solve to good grades and the class ending! MAAATH!”
Entire Class:
“MAAATH!”
Professor:
“MAAAAATH!”
Entire Class:
“MAAAAAATH!”
Professor:
“Forth, exam-takers!”
(The entire class rises to their feet and gives him a standing ovation. A week later, we get an email from the professor.)
Professor:
*at the end of the email* “PS: I appreciate all of you who wrote in their evaluations that I was the one professor to rule them all, but the best one yet was the student who called me ‘Mathrandir.’”

Filed under math puns lord of the rings calculus final

32 notes

secxtanx: theworstmathematician: (This picture is from the book Math Made Visual by Claudi Alsina and Roger B. Nelsen) Given any two circles, there are exactly two lines that are tangent to both those circles at the same time. If the circles are different sizes, then these two tangents cross at some point. If we have three circles, we can build three such points by looking at the common tangents to every pair of circles. Monge’s theorem states that these three intersections always lie on a line. The proof in the book is cute: Let each circle be the “equator” of a sphere. Given a pair of spheres consider the cone generated by the two corresponding tangent lines. Half of the cone will lie above the plane of the circles and half will lie below. Now consider a plane tangent to the three half-spheres. This plane will also be tangent to each of the three cones, and it will intersect the original plane in a line L. Since this plane contains one line from each half-cone, the vertices of the three cones must be located on the intersection line L.  .‿.  cute? downright adorable.  Very nice. I wonder what the projective version of this theorem is? (I.e., if you replace the circles with conic sections.)

secxtanx:

theworstmathematician:

(This picture is from the book Math Made Visual by Claudi Alsina and Roger B. Nelsen)

Given any two circles, there are exactly two lines that are tangent to both those circles at the same time. If the circles are different sizes, then these two tangents cross at some point.


If we have three circles, we can build three such points by looking at the common tangents to every pair of circles.

Monge’s theorem states that these three intersections always lie on a line.

The proof in the book is cute:

Let each circle be the “equator” of a sphere. Given a pair of spheres consider the cone generated by the two corresponding tangent lines. Half of the cone will lie above the plane of the circles and half will lie below. Now consider a plane tangent to the three half-spheres. This plane will also be tangent to each of the three cones, and it will intersect the original plane in a line L. Since this plane contains one line from each half-cone, the vertices of the three cones must be located on the intersection line L.

 .‿. 

cute? downright adorable. 

Very nice. I wonder what the projective version of this theorem is? (I.e., if you replace the circles with conic sections.)

(Source: twocubes)

Filed under math circles tangent Monge's theorem visual projective geometry

20 notes

I came across these curves while writing a test: they are the intersections of the unit sphere with helicoids of decreasing “pitch”. I was surprised to find all the curves are smooth (although, in retrospect, I shouldn’t have been). They are parametrized by 

f(t) = (sin(t)*cos((n*cos(t)), sin(t)*sin(n*cos(t)), cos(t)).

I’ve shown the cases n = 3, 4, 5, 10, 20, 50, 100. The curves clearly become dense on the sphere as n approaches ∞. My question is, do the parametrized curves converge to a sphere-filling curve, i.e., a continuous map of the circle onto the sphere? Probably not, but I thought I’d throw the question out there, particularly since I think the curves are pretty.
 
Additional thoughts: In the above parametrization, n doesn’t have to be an integer. So there are really two questions at play. First, do the curves converge as n approaches ∞ through the reals? Second, is there a sequence of these curves that converges?

Filed under math geometry analysis question

267 notes

1ucasvb: The ballistic ellipse This is something I found when I was playing around with ballistic trajectories. I wondered what shape you would get if you connected all the apex points of all trajectories, if you only changed the angle and kept the same initial speed. Surprisingly, you get an ellipse! EDIT: Also, here it is in 3D! Naturally, you get an ellipsoid. The equation for the ellipse is: x2 / a2 + (y - b)2 / b2 = 1 Where a = v02 / (2g) and b = v02 / (4g). Naturally, v0 is the initial speed and g is the acceleration due to gravity. In another curiosity, the eccentricity of this ellipse is constant for all values of v0 and g, and this value is e = √3 / 2. Obviously, I wasn’t the first to find this. A quick search revealed a paper on arXiv from 2004 describing this. Still, it’s a nice little-known curiosity of a classical physics problem. Bonus points: for which angles does the trajectory contain the foci of the ellipse? Everyone should be following this blog.

1ucasvb:

The ballistic ellipse

This is something I found when I was playing around with ballistic trajectories. I wondered what shape you would get if you connected all the apex points of all trajectories, if you only changed the angle and kept the same initial speed.

Surprisingly, you get an ellipse!

EDIT: Also, here it is in 3D! Naturally, you get an ellipsoid.

The equation for the ellipse is:

x2 / a2 + (y - b)2 / b2 = 1

Where a = v02 / (2g) and b = v02 / (4g). Naturally, v0 is the initial speed and g is the acceleration due to gravity.

In another curiosity, the eccentricity of this ellipse is constant for all values of v0 and g, and this value is e = √3 / 2.

Obviously, I wasn’t the first to find this. A quick search revealed a paper on arXiv from 2004 describing this. Still, it’s a nice little-known curiosity of a classical physics problem.

Bonus points: for which angles does the trajectory contain the foci of the ellipse?

Everyone should be following this blog.

Filed under math ellipse trajectories physics

849 notes

1ucasvb: In a previous post, I showed how to geometrically construct a sine-like function for a regular polygon. I also pointed out how the shape of the function’s graph depends on the orientation of the polygon, since it isn’t perfectly symmetric like the circle. This animation illustrates how the polygonal sine (dark curve) and polygonal cosines (clear curve) change as the generating polygon rotates. Derivation In order to find these functions for an arbitrary polygon, we first need to write the polygon in polar form. That is, we want the radius for a given angle. In a circle, this is a constant value. A general “Polar Polygon” function is: PPn(x) = sec((2/n)·arcsin(sin((n/2)·x))) Where n is the number of sides of the polygon. If n is not an integer, the curve is not closed. Armed with this function, we can quickly find the polygonal sine and polygonal cosine: Psinn(x) = PPn(x)·sin(x)Pcosn(x) = PPn(x)·cos(x) As n grows, the functions approximate the circular ones, as expected. To rotate the polygon, just add an angle offset to the x in PPn. So, what is it good for? I’ve used this several times when I wanted some smooth interpolation between a circle and a polygon, in such a way that the endpoints of the interpolation are a perfect circle and a perfect, pointy polygon. It’s useful in parametric surfaces, such as in this old avatar of mine:

1ucasvb:

In a previous post, I showed how to geometrically construct a sine-like function for a regular polygon.

I also pointed out how the shape of the function’s graph depends on the orientation of the polygon, since it isn’t perfectly symmetric like the circle.

This animation illustrates how the polygonal sine (dark curve) and polygonal cosines (clear curve) change as the generating polygon rotates.

Derivation

In order to find these functions for an arbitrary polygon, we first need to write the polygon in polar form. That is, we want the radius for a given angle. In a circle, this is a constant value.

A general “Polar Polygon” function is:

PPn(x) = sec((2/n)·arcsin(sin((n/2)·x)))

Where n is the number of sides of the polygon. If n is not an integer, the curve is not closed.

Armed with this function, we can quickly find the polygonal sine and polygonal cosine:

Psinn(x) = PPn(x)·sin(x)
Pcosn(x) = PPn(x)·cos(x)

As n grows, the functions approximate the circular ones, as expected. To rotate the polygon, just add an angle offset to the x in PPn.

So, what is it good for?

I’ve used this several times when I wanted some smooth interpolation between a circle and a polygon, in such a way that the endpoints of the interpolation are a perfect circle and a perfect, pointy polygon. It’s useful in parametric surfaces, such as in this old avatar of mine:

(via greentea-strawberries-deactivat)

Filed under math polygons trigonometry gif

12 notes

Found this while looking through my calculus textbook. Maybe if Angela and Brian could bond over their shared love (or hate) of math problems with completely artificial “context”…

Found this while looking through my calculus textbook. Maybe if Angela and Brian could bond over their shared love (or hate) of math problems with completely artificial “context”…

Filed under math calculus funny