# Algebra

## Farewell 2021 with 3n+1

The Youtube algorithm one day pushed a video "The Simplest Math Problem No One Can Solve - Collatz Conjecture" to me. After viewing the video, I thought that my VRMath2 could easily visualise the 3n+1 graph with its LOGO programing. I am happy that my VRMath2 application is still working after almost three years in low maintenance mode. With some simple coding. I have written a LOGO program that can take a number (natural number) and the turtle will draw its 3n+1 graph. The 3n+1 problem is simple. It starts with a natural number; if the number is odd, multiply it by 3 then add one; if the number is even, the number is divided by 2. These rules apply until the resulting number is one. It is interesting that it seems any given natural number will end with 1 after applying the 3n+1 rules, but how?

## Sodium Fluoride

Tis is my introduction

## Ripple wave

This is my first water ripple experiment. I found a function on Youtube and had a quick experiment. The ripple effect is not perfect but could be a good start for whoever is interested to refine it.

## 3D function with ElevationGrid

I have been thinking about 3D graphing using ElevationGrid for a while and finally got time to demonstrate here. Previously, I have blogged about 3D plotting and ElevationGrid, This blog is simply taking the same maths function f(x,z) = x*z^3-z*x^3 in the 3D plotting blog and apply to the ElevationGrid object.

## VAM Temple project

The VAM Temple project was completed by 3 primary school students (Year 5, aged 9) in 2003, using the archived VRMath 1.0 application. The Logo program they wrote at the time can still run in the new VRMath2 Editor with few modifications. You can try to recreate it in the VRMath2 Editor, by openning the  vam_temple.logo in the Logo Editor and executing the program.

## An attempt at using recursion

All of my projects to date have required step by step instructions (some upwards of 300 lines of code!), so I decided to have a go at working out how to create a recursive code. After studying "Cayley graph 3D" working out in my head how the code was constructed, I was finally able to start my own. The following is based on an image of

Cayley Graph of the Free Product Z3 * Z5

After much trial and error I was able to develop ...

## Cayley graph 3D

I was in the keynote session of AAMT conference. In the keynote, the mathematician Hanna Neumann was mentioned. I immediately googled and started reading about her on Wikipedia.

While reading, my thoughts are like the hyperlinks that go everywhere, then suddenly I saw and clicked into the Group theory, where I found the Cayley graph that caught my attention.

It is easily recognisable that this cayley graph is a fractal image, which can be produced with a simple recursive procedure in VRMath2's LOGO language.

## Recursive 2D fern leaf procedure

In my previous blog, I explained about a recursive tree procedure. Here is another one borrowed from Joshua's Logo Interpreter example: fern leaf procedure. This fern leaf procedure creates a 2D fern leaf in VRMath2's 3D space. You can examine it below in the 3D space with the LOGO program.

## Recursive 2D tree

One of the powerful abilities of LOGO programming language is its recursive execution of procedure. Recursive call of procedure means a procedure calls itself until an exiting condition is met. A recursive procedure can be very simple yet it can produce complex and beautiful mathematical graphics such as fractal images and naturally occurring objects.