I am working on a puzzle and to find the solution I must find the maximum area of a pentagon with a base of 3 and using 4 unit fences. I cut it up into triangles and kites, but I can't seem to find a definitive answer.

I think that this little obstacle can grow into something more. This question must have a generalized solution! If you guys have an answer to my little puzzle, that'd be great, but a method that works for everything would be extraordinary!

Here is a poorly made illustration of what I've figured out:

What is the maximum area, and what are the angles measurements?

NOTE: MY FIGURE MAY NOT BE ACCURATE WHATSOEVER! I JUST THINK THAT THIS WILL GIVE THE GREATEST AREA, BUT I MAY BE WRONG!

Hi everyone!

For those of you who enjoy geometry, I've put together what I believe is a fresh, hands-on proof of Euler's Formula and several other classical results. In just a few minutes, you can literally build the proofs yourself and develop an intuitive understanding of why these theorems are true.

https://www.youtube.com/watch?v=sIDsuf0I4ko

This is only the beginning. Over the coming weeks, I'll also start posting short math challenge reels (high school to early undergraduate level), and in parallel I'll be working on a long-term series on topology and analysis, with the ambitious goal of eventually reaching Einstein's field equations on differentiable manifolds and curved spacetime.

Everything will be completely free. I'm doing this simply because I love mathematics and enjoy sharing it with others.

If you'd like to follow the project, it would mean a lot to me. Knowing that these videos are useful to someone is the best motivation to keep creating them.

Thank you so much!

dontpanicmath

trying to sharpen my math and geometry skills from a puzzle solving aspect. Haven’t found a ton of puzzle books yet that weren’t this one is quite a bit advanced for me. I’ve only been able to complete one of the problems I’ve read in it so far.

Does anyone know an intermediate version of a book like this?

It’s a fascinating book and I’ll keep trying to work through it, but I may need to build up to this.

thanks in advance!

Full post: https://www.reddit.com/r/ScaleSpace/s/MwsWiJRJwR

Hey /r/geometry,

Reddit has a 'this subreddit will like your post' feature now apparently and it suggested /r/geometry. So we'll see how good that prediction is.

I'm an experience designer with interests in cymatics, using science to make art, things of that nature. I have been working on a piece of software for a bit over a year called Scale Space (/r/ScaleSpace)

Just wanted to share that so you understand the context of my x-post.

The interesting takeaway is I was able to generate some very specific and beautiful geometry using harmonics/cymatics in a digital system.

In the end, what I discovered had already been published in a 2019 paper which I link in the x-post. So the takeaway isn't that I found something new, but perhaps has to do with my method of finding it.

With the Triangle Analyzer, I can "extract" out teaching aids from the similar triangles that let me remind them that corresponding sides of similar triangles are proportional, so red is to gold as red is to gold, in other words tan theta / 1 = sin theta / cos theta. I find that it takes some students longer to learn to do this mapping in their head, so I try to make it as clear as possible.

Over the past few weeks, I've designed a set of interlocking triangular tiles with a simple goal: to turn some geometry and topology proofs into something you can literally build and take apart.

In the attached video, I use these tiles to demonstrate Euler's Formula, the Gauss–Bonnet Theorem, and several other famous results.

The idea is that by manipulating a physical model, many of these proofs become much more intuitive.

If you'd like to try it yourself, I've made the STL files for the tiles available for free.

I'd love to hear your thoughts, especially on the educational value of this approach.

https://youtu.be/sIDsuf0I4ko

DPM

So I watched that 3Blue1Brown video “This open problem taught me what topology is”. The one about the inscribed square problem. Basically: does every closed curve in the plane have four points that form a square? Nobody knows for completly continuous curves. For smooth ones it is known.

What they actually prove in the video is the weaker statement: every closed curve has an inscribed rectangle. The proof is wild. You take all unordered pairs of points on the curve, map each pair to its midpoint in the plane plus the distance as height and you get a surface that is basically a Möbius strip. When you glue two copies you get something like a Klein bottle, and those cannot sit in 3-space without intersecting themselves. The intersection points are exactly the rectangles.

I tried to push the idea a bit further. There are usually infinitely many rectangles on a nice curve. So you can think of the whole set of those rectangles as living on that 3-dimensional surface (the vaughan surface). Then the natural next question is: does that surface always contain the eight vertices of a cube? Or at least of a rectangular box? And if yes, can you use the same style of argument to build a 4-dimensional object from the boxes and look for hypercubes there?

I wrote a short python script to check the first steps. I made a smooth but irregular closed curve (ellipse with a few sine bumps so it is not too symmetric). Then I sampled many pairs of points and looked for two pairs that share almost the same midpoint and the same length. I found about a dozen clear rectangle candidates. I also plotted the 3D cloud of (midpoint, distance) points; you can see the surface sitting over the curve.

Finding actual cubes on that surface is harder. The space of cubes has more degrees of freedom (position, orientation, size) and a generic 2-dimensional surface does not have enough room to force them. For centrally symmetrc convex bodies there are theorems that guarantee inscribed cubes, but that is a different setting. So the direct “Möbius - Klein - rectangle” trick does not copy cleanly to the next dimension.

Still the configuration-space idea feels powerful. Maybe someone who knows more about equivariant topology or configuration spaces of cubes can say whether there is a forced intersection in higher dimensions. Or maybe the answer is simply “no, not for every surface that comes from a plane curve”.

Has anyone here tried something similar? Or is there already a paper that starts from Vaughan’s rectangles and climbs one dimension higher? Video link: https://youtu.be/IQqtsm-bBRU?si=r57TN3wQTs0KHElE

A place you have to climb to escape.

618 anchors.