Geometric Bites

The rules of L-systems can be used to generate shapes such as the pentagon . The basic rules of L-systems are: F : move forward a distance - : change direction and go clockwise + : change direction and go anti-clockwise Since a pentagon has external angles of 72 degrees (as shown in the workings below), when using an L-system, you'd have to set the turn angle to that figure to generate the desired outcome. So, to draw a pentagon, you'd need to use the rule: F+F+F+F+F [Turn angle: 72 degrees] And that would give you the correct drawing. I learnt this method in the book 'Pattern Generation for Computational Art' by Stefan Hollos and J. Richard Hollos. It contains very good tutorials about how to apply L-systems, so I would recommend it to anyone interested in this kind of computational art. You can purchase this book via my affiliate link here: https://amzn.to/3DQe5Iw Purchases made through my affiliate links help sustain this blog and my maths project, so many thanks...

For at least a decade I have been using the Helix Oxford 12 inch 30cm Shatter Resistant Ruler . It's a product I recommend because it's affordable and flexible , and it will generally last for a long time, that is, providing you don't use it for stencil work and you keep it clean. Below is my video recommendation... If you are interested in buying this ruler, my affiliate link is below: https://amzn.to/3zYUMKw By making a purchase through my affiliate links, you help support my blog and Youtube channel. Many thanks!

Below are the workings you can use to find the intersection points of two quadratic equations . Part 1 Part 2   There is a free interactive graph related to these workings here: https://www.desmos.com/calculator/0twcp4hwx2

Today I was watching a video about how to construct a dodecahedron . In the first part of the clip, it was shown how to get the golden ratio with a specific type of geometric construction . I was absolutely intrigued by it and decided to figure out if the construction did indeed yield the result phi. Remarkably and beautifully, I was able to confirm the result. My workings are shown below. I live for these eureka moments! Part 1 Part 2 Part 3   Part 4

In the workings below I demonstrate how you can derive 1+tan²(θ)=sec²(θ) and 1+cot²(θ)=cosec²(θ) using sin²(θ)+cos²(θ)=1 .   Just follow the instructions, and presto!

Below I demonstrate how to derive tan(π/8) , sin(π/8) and cos(π/8) from scratch. These workings are a bit similar to the workings for tan((3π)/8) , sin((3π)/8) and cos((3π)/8) which can be found here : https://geometricbites.blogspot.com/2021/08/how-to-derive-tan38-sin38-and-cos38.html The edge with length R √(2- √(2)) was actually derived in the tan((3π)/8), sin((3π)/8) and cos((3π)/8) post, so doesn't need to be found again. Part 1  Part 2  Part 3  Part 4