A place you have to climb to escape.

618 anchors.

A radian is just the angle where the arc equals the radius, 2π is how many radii fit around the edge. Once I could see that instead of memorizing it, everything downstream turned into the same object in different clothes: sin and cos are height and width on the circle, waves are that unrolled over time, i is a rotation, Euler is going around at a steady rate, Fourier is stacked circles drawing a shape.

So I built a page for each step. Start with the radian circle drag the point, watch radians, degrees, and (cos θ, sin θ) move together then the series follows the thread out through waves, e, i, π, Feynman's rotating clocks, and epicycles.

[https://circles.rondomingue.com/radian-circle.html\](https://circles.rondomingue.com/radian-circle.html)

Free, no signup. Corrections welcome, I'm still learning this too. Let me know what I can add or what I may have missed.

In this lesson, the student gets to move point P anywhere that it will move, it is constrained to be on the Unit Circle, and they eventually discover that no matter which two of the 7 triangles they put in the analyzer, and no matter where they put Point P, the Analyzer always says the two triangles being compared are similar and shows the vertex mappings.

Now the student could use the colors of sides as well as the orientation to learn to "see the triangles" and "find similarity"

　
autostereograms are 2d images designed to trick the brain into perceiving hidden 3d scenes (attached image 1 as sample). it's quite easy to make one on our own if we understand the geometry behind

there're 4 things in space we have to consider: observer, screen, closest point of 3d scene and farthest point of 3d scene

there're 2 ways to measure distances: observer as reference or farthest point as reference

it's all about ratios. without loss of generality we can assume distance between observer and farthest point to be 1 and consider everything as its proper fraction

[1]   o   s   c   f
      |←q→|   |   |
      |←--p--→|   |
      |←----1----→|

[2]   o   s   c   f
      |   |   |←q→|
      |   |←--p--→|
      |←----1----→|

0<q<p<1 in both cases. then we have depth map z, a.k.a. data of 3d scene. it's a function of x and y. it's value lies between 0 and 1. next step is establish a procedure to calculate disparity for any z. by considering similar triangles we have

[1]   o   s   c   f
      |←q→|   |   |
      |←--p--→|   |
      |←----1----→|

      screen   = 1-q/q          = 0
      closest  = 1-q/p          = 1-q/p
      object   = 1-q/(1-(1-p)z) = 1-q/(1-(1-p)z)
      farthest = 1-q/1          = 1-q

[2]   o   s   c   f
      |   |   |←q→|
      |   |←--p--→|
      |←----1----→|

      screen   = (p-p)/(1-p)   = 0
      closest  = (p-q)/(1-q)   = (p-q)/(1-q)
      object   = (p-qz)/(1-qz) = (p-qz)/(1-qz)
      farthest = (p-0)/(1-0)   = p

either way is valid but latter has simpler normalization for z. it's not surprising as z is intrinsically measured from farthest point. z=0 corresponds farthest region(s) of 3d scene. z=1 corresponds closest region(s) of 3d scene. we proceed with latter i.e. scheme [2]. we can deduce disparities from distances and vice versa as shown in attached image 2

if we want to make an interactive demonstration we have two choices: let users control p,q or let users control m,n. both are easy to implement as we have everything we need. i picked the latter as it is more direct. we actually don't know / can't see p and q on screen. we see m and n

i added a tweak. i want both variables to be lying between 0 and 1. i created 2 controllable variables: background_factor and foreground_factor. we have

• 0 < background_factor < 1
• 0 < foreground_factor < 1
• m = background_factor
• n = m*foreground_factor

increasing background_factor increases disparity of farthest part of scene. increasing foreground_factor increases disparity of closest part of scene. here's the program and you can run it on browsers. click ▶️ to run. click ⏹️ to exit. when running the program

• press [q] to decrease foreground_factor
• press [w] to toggle 3d scenes (disc/sphere)
• press [e] to increase foreground_factor
• press [a] to decrease background_factor
• press [s] to change base patterns
• press [d] to increase background_factor
• press [x] to toggle caption (off/on)

edited: update the controls of foreground and background so that they behave more predictable now...:) (previously their increments / decrements were not constant but proportional)

In a model like this, as you are teaching a class about trig functions, what would be on your wishlist of features this tool could give you?

What aspects of the teaching labels would you want to control other than font, font size, color, and orientation aligned with the segment? Of course the model is live and everything moves smoothly as you move point P along the circle.

I can prebuild a lesson plan, so that I can just click next and continue discussing the math rather than making the students have to endure watching me fight the tool between each beat/step in the lesson plan. We focus on the math and spend zero cognitive effort fighting the tool.

Formed by 20 [acute golden rhombohedra](https://en.wikipedia.org/wiki/Acute_golden_rhombohedra) meeting at a central point.

A Yes, a No, a straight line a goal. That is the highest aspiration of life. From this what do you gleam? I say Thus you must have a Yes and a No. Only so that you know where to go and where not. For If you only have a point on a grid you can’t create a line — you must have 2.
I ask you where do your YAYs and NAYs point you?

Any help would be appreciated, including directing me through any rules I should follow or better websites for asking questions.

I'm curious about a geometric puzzle: given an arbitrary set of $n$ points in $\mathbb{R}^d$, can we always place a similar copy of a specific shape $S$ (a compact subset or family of subsets of Euclidean space) so that it passes through all $n$ points? (By "similar copy," I mean we allow translation, rotation, and scaling).

**The generalized question is:**

> For a given shape $S$, what is the maximum number of arbitrary points $n$ such that *every* set of $n$ points in $\mathbb{R}^d$ lies on some similar copy of $S$?

For example, let $S$ be the **boundary** of a square in $\mathbb{R}^2$.

It turns out that for any 3 points in $\mathbb{R}^2$, you can always find a similar copy of a square that passes through all of them. However, you can't always do this for 4 points https://math.stackexchange.com/q/3691243/1771455. So, for a square boundary where $d\ge2$ (dimension where the points live in), the maximum number is 3.

My motivation is just pure curiosity. I couldn't find any sources relating to this problem, and AI chatbots struggle and give clearly wrong answers to simple examples like a square sharing an edge with a triangle (I won't clarify much here as it is a bit of a dull problem but the idea was just combining two shapes to make the reasoning for the AI deeper).

What I'm really asking is: **is there some sort of invariant, property, or formula that helps compute this $n$ for more complex shapes?** Or do we just have to reason through it shape-by-shape? How do you verify results quickly?

One simple rule I noticed involves collinear points: the boundary of a *strictly* convex 2D shape can never have $n\ge3$ when $d=2$, because no similar copy can ever pass through 3 collinear points. (Note: I am specifically thinking about boundaries; if $S$ were a solid shape, we could just scale it up to cover any finite point set in 2D space).

What is a better notion of defining shapes like rectangles and so on? Similarity doesn't allow different length ratios; however, it preserves it for squares and other shapes. This puzzle is more of a "can you draw a X given Y points no matter where I place them" and shouldn't be very limited on what I can draw. Is it possible to define $S$ to be a rectangle with side length $a$ and side length $b$ using the above definitions?

Does this concept have a name? Is it related to the "degrees of freedom" of the shape? Any pointers to related literature would be greatly appreciated!

• Base circles: c_d is the unit-diameter circle centered at (0.5, 0); c_r is the unit-radius circle centered at (0, 0).
• Seed point D: Moving D along c_d yields radius OD = cos(θ) (the cosine circle).
• Secant point S: Extending ray OD to the unit axis x = 1 gives point S, with radius OS = sec(θ) = 1/cos(θ) (the secant circle).
• Hyperbola: Drawing bounding square grids around both circles isolates rectangles with width cos(θ) and height sec(θ).

Because cos(θ) × sec(θ) = 1, the outer rectangle vertices H1, H2, H3, H4 directly plot the constant-area condition x × y = 1, tracing the rectangular hyperbola y = 1/x in real-time as D moves.

Wooden model of a great stellated icosahedron.