Geometric Bites

Here is my video recommendation of the Sharpie S Gel 0.7 pen. I think it's fantastic for maths. Not only does it look good, it's light, comfortable and doesn't bleed. I thoroughly recommend it. If you are looking for a pen to write proofs with or solve problems, you're going to love this product. You can buy this pen via my Amazon affiliate link here: Click here to buy the Sharpie S Gel 0.7 on Amazon Purchasing through my affiliate links is a way to support my blog.

Below I demonstrate how to derive tan((3π)/8) , sin((3π)/8) and cos((3π)/8) from scratch. I use geometry and also algebra . Part 1  Part 2 Part 3 Part 4 Part 5  Part 6  If you'd like quick updates every time I post, follow me on Twitter at: https://twitter.com/tiago_hands .

Here I review the STAEDTLER 552 01 PR1 Mars pair of compasses . I have been using these for the last few years and I thoroughly recommend them. You can find work that has been produced with them at https://www.instagram.com/mathematics.proofs and also https://www.instagram.com/tiago_hands . If you purchase this pair of compasses via my Amazon affiliate link, not only will you get a fantastic item, but at the same time, you'd be supporting this blog and my Youtube channel. Buy the STAEDTLER 552 01 PR1 Mars Comfort compass set on Amazon Thank you!

Below I demonstrate how to find the line that passes through two intersecting circles. Part 1 Part 2 Part 3   There is a free interactive graph that shows why this is correct. https://www.desmos.com/calculator/z9rh2ri9bz

Here is a video review of the Linkstyle Electric Pencil Sharpener I recently bought from Amazon. I use an electric sharpener so that I can create my geometric drawings more quickly, which is why I think this is a worthy recommendation. You can find this product via:  https://amzn.to/2V3teol *I may receive referral fees from my product recommendations on Amazon. I recommend the products I feel are best for me.

Below is an example of how to find the region where two circles on a 2 dimensional plane overlap . The workings are in high resolution... To demonstrate the workings are indeed correct, a free graph has been created on Desmos: https://www.desmos.com/calculator/0aonrev54r

In this post I demonstrate how to find out where a line and circle intersect on a 2 dimensional plane. The first step is to write out the equations of a line and a circle. We then make sure for both equations 'y' is isolated . Once we have the equations for a line and a circle whereby 'y' is isolated, we can then go about finding the values of 'x' for where the line intersects the circle. Like so... After we have found the values of 'x' using the quadratic formula , we then plug them back in to y=mx+c to get the values of 'y' for which the line intersects the circle. And that's it basically. Below is a free interactive Desmos graph related to this work: https://www.desmos.com/calculator/w3tsnjajtx

Below are the workings required to build an interactive graph that finds the points where a line and quadratic equation intersect . Here 'a' is not equal to 0 as it becomes part of the denominator a fraction . Also, if 'a' were equal to 0, we'd have a line intersecting a line, not a parabola (for instance). When we have the formula for 'x', we just plug it back into the linear equation to get the outcomes for 'y'. And there we have it, the points where a line and quadratic equation intersect. To see the interactive graph I was talking about, visit the Desmos link below: https://www.desmos.com/calculator/sqzjajbhea

Recently, I produced some recursive geometric art that got a pretty good amount of attention on the internet. Here it is: So, what was the secret of its success? It was a solid geometric drawing I made beforehand. With that solid geometric drawing, I was able to see the proportions required to produce my animation. There is a clarity that geometric drawings bring because of their rigidity . As you can see in the image above, the squares get smaller . Their lengths decrease by a factor of 1/2 with each iteration . When these lengths are applied to vertical and horizontal lines what you get is pure mathematical magic... The pattern in my video. How about the equations ? Well, if I tell you everything it would be like revealing the entire story of a great movie before you've watched it. I would hate to spoil your creative experience. :-)

This proof solves a popular internet geometry problem found here: https://twitter.com/RonySarker71/status/1428469920443076609 By utilising the radius of the circle , one can arrive at the conclusion that y= √(ax). If a=1, then you get y= √(x).  

A square the quarter the size of another square will have a radius half the size. I show this in my workings below.   Why is this useful? Well, this truth can be used to create recursive square art using graphing apps like Desmos . In one of my next posts you'll see exactly what I mean. I have some art already planned. Stay tuned!