Plot of the function f(n) = sin(1) + sin(2) + ... + sin(n)
For a positive integer n, let f(n) = sin(1) + sin(2) + ... + sin(n), where sin is measured in radians. Plotting the values of f gives us a graph with some interesting structure.
www.desmos.com/calculator/l...
As a challenge, try to find the best bounds on f.
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Origami sculpture of five intersecting tetrahedra
Here is the origami sculpture, and folding instructions:
steemit.com/origami/@iva...
I was gifted a gorgeous origami sculpture of five intersecting tetrahedra. This inspired me to recreate it using Desmos.
www.desmos.com/3d/m7hqyu2gpr
en.wikipedia.org/wiki/Compoun...
@desmos.com #iTeachMath
An ellipse rotates so that it stays in contact with the coordinate axes. What curve does the center of the ellipse trace? (If you are feeling ambitious, you can try finding the curve the foci trace, and its area.)
www.desmos.com/calculator/v...
#iTeachMath @desmos.com
Hence, (10P - 3A)/7 = (4O + 3B)/7. The coefficients on both sides add up to 1, so this common vector lies on both lines AP and OB. Therefore, the common vector coincides with point N, which means N = (4O + 3B)/7. This gives us ON:NB = 3:4.
From the given, M = (A + B)/2 and P = (2O + 3M)/5. Substituting, we get P = (2O + 3(A + B)/2)/5 = (4O + 3A + 3B)/10. Then 10P = 4O + 3A + 3B, so 10P - 3A = 4O + 3B.
Similarly, a = -1 and b = 6 implies a^2 + ab + b^2 = 31, and a = 1 and b = 5 implies a^2 + ab + b^2 = 31. This suggests that taking a or b negative does not give us any additional values.
That's an interesting point. For example, if we take a = -3 and b = 5, then we get a^2 + ab + b^2 = (-3)^2 + (-3)(5) + 5^2 = 19. But if a = 2 and b = 3, then a^2 + ab + b^2 = 2^2 + (2)(3) + 3^2 = 19.
Your table leaves out the values of a^2 + ab + b^2 where a or b is 0 (which are the perfect squares).
mathworld.wolfram.com/BrianchonPoi...
en.wikipedia.org/wiki/Triangl...
Given triangle ABC and point P, let AP, BP, CP intersect BC, AC, AB at D, E, F. Then there exists a unique conic that is tangent to the sides of triangle at D, E, F (known as an inconic). The point P is known as the Brianchon point (also perspector) of the inconic.
This ties nicely to your recent posts on difference of squares.
We can write 91 = 100 - 9. Then by difference of squares, 100 - 9 = (10 - 3)(10 + 3) = 7*13.
Ad with picture of woman who is struggling with addition.
Some people need more help than others. #iTeachMath
Triangle ABC with squares AB B1 B2 and AC C1 C2 on sides AB and AC. M is the midpoint of B1 and C1.
Given triangle ABC, we construct squares AB B1 B2 and AC C1 C2 on the sides. Let M be the midpoint of B1 and C1. As point A varies, what do you notice about the position of point M?
www.desmos.com/calculator/3...
@desmos.com #iTeachMath
Diagram of tetrahedron, octahedron, and cube in spheres.
Suppose we want to arrange N points on a sphere, so that the smallest distance between any two points is maximized. For N = 4, 6, 8, we can take a regular tetrahedron, regular octahedron, and cube, respectively.
One of these is not the optimal solution. Which one?
#iTeachMath
Geometric diagram and its inverse.
As @diffgeom.bsky.social points out, inversion takes care of the problem nicely. (And many of these Sangaku problems were solved by inversion.)
The three arcs invert to three lines. There are two circles that are tangent to these three lines; we want the "top" one.
Here is a graph of a regular 2n-gon using absolute value. (You can adjust the value of n using the slider.) www.desmos.com/calculator/c...
I have confirmed that your answer is correct.
Photo of tire hubcaps with different kinds of symmetry.
While out on a walk, I noticed that the hubcaps on tires have very different kinds of symmetry. What kind of symmetry does your car have? What about the cars around you? #iTeachMath
The model of Dandelin spheres works for cones and cylinders: en.wikipedia.org/wiki/Dandeli...
Table of Lissajous Curves
Table of Lissajous Curves www.desmos.com/calculator/z...
I've seen this animation online, so I thought I'd try building it myself.
@desmos.com #iTeachMath
Trig is like an acquaintance who introduces themselves as being about triangles.
When you really get to know them, their real identity is uncovered: they are about circles (more specifically about triangles inside circles w/base a radius from center edge)
#iTeachMath
In the coordinate plane, A = (a,0), B = (0,b), and C = (p,q).
In the diagram below, triangle ABC is equilateral.
Given a and b, can you find p and q?
Given p and q, can you find a and b?
#iTeachMath
Nice! This is known as the Leaning tower illusion (en.wikipedia.org/wiki/Leaning...)
I have used an example like this, to illustrate to students that you should not trust geometric diagrams too much, e.g. lines that look parallel may not actually be parallel.
Three congruent circles are inscribed in a triangle.
Three congruent circles are inscribed in a triangle, as shown below. Can you compute the common radius? www.desmos.com/calculator/p... @desmos.com #iTeachMath
This is a consequence of the Incenter-Excenter lemma (web.evanchen.cc/handouts/Fac...): In triangle ABC, if I is the incenter, I_A is the A-excenter, and M is the midpoint of arc BC, then MB = MC = MI = MI_A.
Glad to help!
Here is a graph: www.desmos.com/calculator/m...
Points O, X, and Y are draggable. There is a slider for n, the number of lines. @desmos.com
Great activity!
Slide 12 says "Move the red point so that it is equidistant from the areas of the triangle." Should it say something about the areas of the triangles being equal?
