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nsato7.bsky.social

@nsato7.bsky.social
36 followers 49 following 59 posts

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nsato7.bsky.social @nsato7.bsky.social · 27/09/2026
Rhombic Tiling of a Regular Polygon www.desmos.com/calculator/y... #iTeachMath @desmos.com
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nsato7.bsky.social @nsato7.bsky.social · 17/09/2026
Suppose you take k points in the interval [0,2π] (spaced apart by 2π/k) and map them to the rose curve r = cos nθ. Then you can get interesting looking graphs - and sometimes, a regular polygon. (Thanks to @helsig.bsky.social for the idea.) www.desmos.com/calculator/g... #iTeachMath @desmos.com
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nsato7.bsky.social @nsato7.bsky.social · 13/09/2026
This means you can "stack" the hypocycloids: www.desmos.com/calculator/l... www.desmos.com/calculator/b...
desmos.com
General Nested Hypocycloids
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nsato7.bsky.social @nsato7.bsky.social · 11/09/2026
I posted the link on aops; I can take it down.
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nsato7.bsky.social @nsato7.bsky.social · 11/09/2026
At 10:30, you derive that the x-coordinate a' is (ab - 1)/(b - a). But it should be (b - a)/(ab - 1). Here is a Desmos graph which confirms it: www.desmos.com/calculator/v...
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nsato7.bsky.social @nsato7.bsky.social · 29/08/2026
How does this work?  Can you come up with graphs for other tests of divisibility? (Source: blog.tanyakhovanova.com/2009/08/divi...)
blog.tanyakhovanova.com
Tanya Khovanova's Math Blog » Blog Archive » Divisibility by 7 is a Walk on a Graph, by David Wilson
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nsato7.bsky.social @nsato7.bsky.social · 29/08/2026
For example, for n = 224, move along two black arrows, then a white arrow, then two black arrows, then a white arrow, then four black arrows. You will end up at the bottom node if and only if n is divisible by 7 (like 224).
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nsato7.bsky.social @nsato7.bsky.social · 29/08/2026
You can test whether a number n is divisible by 7 using the attached graph. Start at the bottom node.  For each digit d in n (starting with the left digit), move along the black arrows d times.  Each time you go to the next digit, move along a white arrow. #iTeachMath
Graph with 7 nodes, and black and white arrows.
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nsato7.bsky.social @nsato7.bsky.social · 20/07/2026
The mathematics of soccer balls is highly underrated: www.scientificamerican.com/article/the-...
scientificamerican.com
The Surprising Math and Physics behind the 2026 World Cup Soccer Ball
Here’s how the new tetrahedron-based design for the “Trionda” soccer ball may affect next year’s big game
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nsato7.bsky.social @nsato7.bsky.social · 03/07/2026
I proved a theorem about a quadrilateral which turned out to be obvious. You could say it was a 4-gon conclusion. #iTeachMath
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nsato7.bsky.social @nsato7.bsky.social · 29/06/2026
I have been thinking about Mobius transformations, and how to visualize them. Using Desmos Notebook, I can connect their 2D and 3D representations. www.desmos.com/notebook/ssd... @desmos.com #iTeachMath
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nsato7.bsky.social @nsato7.bsky.social · 23/05/2026
From www.boredpanda.com/funny-textbo...: The Patrick Star graph! #iTeachMath
Graph of z = x^4 + y^4 - 4xy + 1 extended to Patrick Star.
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nsato7.bsky.social @nsato7.bsky.social · 22/05/2026
You might be interested in the hinge theorem: en.wikipedia.org/wiki/Hinge_t... proofwiki.org/wiki/Hinge_T...
en.wikipedia.org
Hinge theorem - Wikipedia
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nsato7.bsky.social @nsato7.bsky.social · 15/05/2026
Interference Pattern www.desmos.com/calculator/g... @desmos.com #iTeachMath
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nsato7.bsky.social @nsato7.bsky.social · 12/05/2026
For an ellipse, it's clear which part is the "inside" and which part is the "outside". For a hyperbola, which part is the "inside"? Which part is the "outside"? #iTeachMath
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nsato7.bsky.social @nsato7.bsky.social · 15/04/2026
I took a topology exam once. It was both open-book and closed-book.
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nsato7.bsky.social @nsato7.bsky.social · 25/03/2026
Is the converse true? That is, if we take a number where the sum of the digits is 9 and divide it by 9, do you always get a number where the digits are increasing from left to right?
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nsato7.bsky.social @nsato7.bsky.social · 25/03/2026
Take a number where the digits are increasing from left to right, and digits can repeat, like 134446689. If we multiply it by 9, then we get 1210020201. The sum of the digits of the new number is 1 + 2 + 1 + 0 + 0 + 2 + 0 + 2 + 0 + 1 = 9. Does this always happen? #iTeachMath
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nsato7.bsky.social @nsato7.bsky.social · 20/02/2026
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. @desmos.com #iTeachMath
Plot of the function f(n) = sin(1) + sin(2) + ... + sin(n)
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nsato7.bsky.social @nsato7.bsky.social · 16/02/2026
Here is the origami sculpture, and folding instructions: steemit.com/origami/@iva...
Origami sculpture of five intersecting tetrahedra
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nsato7.bsky.social @nsato7.bsky.social · 16/02/2026
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
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nsato7.bsky.social @nsato7.bsky.social · 26/01/2026
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
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nsato7.bsky.social @nsato7.bsky.social · 23/01/2026
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.
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nsato7.bsky.social @nsato7.bsky.social · 23/01/2026
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.
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nsato7.bsky.social @nsato7.bsky.social · 21/01/2026
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.
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nsato7.bsky.social @nsato7.bsky.social · 21/01/2026
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.
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nsato7.bsky.social @nsato7.bsky.social · 19/01/2026
Your table leaves out the values of a^2 + ab + b^2 where a or b is 0 (which are the perfect squares).
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nsato7.bsky.social @nsato7.bsky.social · 13/01/2026
mathworld.wolfram.com/BrianchonPoi... en.wikipedia.org/wiki/Triangl...
mathworld.wolfram.com
Brianchon Point -- from Wolfram MathWorld
The point of concurrence of the joins of the vertices of a triangle and the points of contact of an inconic of the triangle (Veblen and Young 1938, p. 111; Eddy and Fritsch 1994). The Brianchon point ...
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nsato7.bsky.social @nsato7.bsky.social · 13/01/2026
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.
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nsato7.bsky.social @nsato7.bsky.social · 11/01/2026
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.
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nsato7.bsky.social @nsato7.bsky.social · 20/11/2025
Some people need more help than others. #iTeachMath
Ad with picture of woman who is struggling with addition.
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nsato7.bsky.social @nsato7.bsky.social · 25/10/2025
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
Triangle ABC with squares AB B1 B2 and AC C1 C2 on sides AB and AC.  M is the midpoint of B1 and C1.
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nsato7.bsky.social @nsato7.bsky.social · 13/10/2025
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
Diagram of tetrahedron, octahedron, and cube in spheres.
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nsato7.bsky.social @nsato7.bsky.social · 05/08/2025
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.
Geometric diagram and its inverse.
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nsato7.bsky.social @nsato7.bsky.social · 21/07/2025
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...
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nsato7.bsky.social @nsato7.bsky.social · 21/06/2025
I have confirmed that your answer is correct.
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nsato7.bsky.social @nsato7.bsky.social · 19/05/2025
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
Photo of tire hubcaps with different kinds of symmetry.
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nsato7.bsky.social @nsato7.bsky.social · 13/03/2025
The model of Dandelin spheres works for cones and cylinders: en.wikipedia.org/wiki/Dandeli...
en.wikipedia.org
Dandelin spheres - Wikipedia
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nsato7.bsky.social @nsato7.bsky.social · 09/03/2025
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
Table of Lissajous Curves
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Reposted by @nsato7.bsky.social
Karen Campe @karencampe.bsky.social · 08/03/2025
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
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nsato7.bsky.social @nsato7.bsky.social · 05/03/2025
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
In the coordinate plane, A = (a,0), B = (0,b), and C = (p,q).
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nsato7.bsky.social @nsato7.bsky.social · 25/02/2025
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.
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nsato7.bsky.social @nsato7.bsky.social · 23/02/2025
I found this: www.youtube.com/watch?v=LjjA...
youtube.com
A Packing Puzzle Requiring Some Math Skills
YouTube video by Dom Puzzles
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nsato7.bsky.social @nsato7.bsky.social · 22/02/2025
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
Three congruent circles are inscribed in a triangle.
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nsato7.bsky.social @nsato7.bsky.social · 21/02/2025
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.
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nsato7.bsky.social @nsato7.bsky.social · 12/02/2025
Glad to help!
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nsato7.bsky.social @nsato7.bsky.social · 11/02/2025
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
desmos.com
Parabola as an Envelope
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nsato7.bsky.social @nsato7.bsky.social · 06/02/2025
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?
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nsato7.bsky.social @nsato7.bsky.social · 29/01/2025
Is this it? ponderingplanning.wordpress.com/2022/10/15/t...
ponderingplanning.wordpress.com
Thinking About Pythagoras’ Theorem
Pythagoras’ Theorem seems to be a concept that can be taught in an endless number of different ways!  I must have re-invented the way that I’ve taught this topic several times.  I’ve also…
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nsato7.bsky.social @nsato7.bsky.social · 01/01/2025
What is the sum of all the numbers in the 9 times 9 table below? #iTeachMath
9 times 9 table
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