## HTML test, & some more maths stuff

Waking up the land’s people

We are going to be covering the basic theory of combinations and permutations that we leaned in this article: http://bit.ly/j918Lc, http://pear.ly/R4cj.

We are also going into some new theory so good luck and I hope you will enjoy this work about maths and HTML code.

rgb(244,000,111) may on occasions be converted here hexadecimal code (eg. #ff9b37).

I have also listed a few maths symbols that are of interest in this particular HTML article:

### Introduction to Group Theory

First of all one of the most famous examples of group theory comes from no other then the Futurama animation, comic & video game: Futurama Writer Invented A New Math Theorem Just To Use In The Show.

cube edges (1,2,3,4)≅ bells (1,2,3,4); the code for this is & # 8 7 7 3;

In school I did not focus on this theory (if my memory serves me correctly), instead I focused on combinations and permutations. So that style of maths will also be mentioned here. Group theory seems much more general (and so more difficult to understand but also more powerful) then permutations and combinations.

### Example of a Isomorphic Musical Gig and the Rotation of a Cube

In abstract algebra Isomorphism is the mapping between two objects. This mapping ends up showing a relationship between two properties or operations.

In essence the below example shows an Isomorphic relationship between two objects (a musical instrument and a cube) by severely restricting the way a cube can move.

We find a relationship between the way two sets are re-arranged even though they do not seem to be alike in any way in our physical world because one is a sound and the other is a visual object. In the world of abstract maths we would believe these things to be exactly same for this particular branch of maths (group theory). However to achieve this result we have to place a lot of rules on how the cube behaves (programming) so it’s patterns both make sense and can try and follow the music in a symbolic way!

The only thing the teacher here misses out on here is rotating the cube colors in 90º intervals (matching the bell colors) in time to the music–very disappointing!

### The possible combinations of 4 bells:

Bells: any of the 4 bells can ring first, then three bells, then two bells then the final bell: 4⋅3⋅2⋅1= 24

### The possible combinations of the orientation of the 6 faces a cube:

Cube: he can show us any of the 6 faces of the cube and each face has the 4 possible orientations (if the rotation of that face is restricted to 90º): 6⋅4= 24; the code for degrees is & # 1 8 6 ;

We also add in a additional rule: While the cube does have 8 different corners, we will be more interested in the square face of a cube and it’s orientation. Thus we could have labeled that faces 1 to 6 and then rotated them four times by 90º to show all the possible orientations of that face.

The teacher has chosen to decorate the cubes faces with 4 identifying colors. The cube’s faces restrict the how many of the 4 color combinations (24) that can be placed on the cube down to only 6 combinations.

Further the corners are fixed and are only able to be rotated (by 90º at a time). Thus each face will store four combinations of the possible 24 combinations of the four colors: Start 0º, 90º, 180º and 270º (rotation to 360º is the same as 0º position). Thus the cube has six sides that can each be rotated into 4 positions: 6⋅4= 24 and so all the four color combinations have been accounted for. However how on earth do we figure out what will be the unique combination of colors that will identify each face.

There probably is a theory for this using circular arrangements, but I’m not going to search for that theory here; I will simply stick to solving the problem by selecting four colors and then rotating the face to get rid of 4 combinations at a time:

let {1,2,3,4} = {red, orange, yellow, green}

```Here a single face of the cube is shown as it is rotated
into it's four possible positions
0º=|1,2|; 90º= |4,1| 180º= |3,4| 270º= |2,3|
|3,4|       |3,2|       |2,1|       |1,4|```
• Four colors on Face 1 (the actual rotation of the faces is shown in the above example):
• 1,2
• 4,3
• Four colors on Face 2:
• 1,3
• 2,4
• Four colors on Face 3:
• 1,4
• 3,2
• Four colors on Face 4:
• 1,2
• 3,4
• Four colors on Face 5:
• 1,4
• 2,3
• Four colors on Face 6:
• 1,3
• 4,2

To make the cube look nice, the teacher has represented the colors by 8 blocks. Obviously to get the 4 colors only, the teacher had to give each color two blocks. Because each face must represent four different colors, there is only one way to place the colors on this occasion–see the video for how that one was solved.

If the combinations of edge numbers (colors on the cube faces in the video) can be made to match the combinations of the ringing bells (the colors also represent one of the four bells–each bell makes a different sound) then we can say that the spinning of the cube edges in a certain way is Isomorphic to the way the bells are played.

While all that seems almost impossible to understand, when you hear and see it in three dimensions the problem seems a lot easier to understand. The simple statement below summarizes much of the evidence (proof) I have provided above.

Isomorphic in simple language means is similar to–but in maths it comes to have a very exact definition and often things that like sounds and cubes in maths are considered similar under certain circumstances–as the above example illustrates!

### C is approximately equal to A/2

C≈ A/2; the code for this is & # 8 7 7 6 ;

C~A/2

### Symbol for multiplication A times C equals D

D = A⋅C; they code for this is & # 8 9 0 1 ;

### Calculating current and power use

R= 100 Ω; code for Ω is & # 9 3 7 ;

Voltage = 12V

I=V/R=12/100=0.12 A

P=V⋅I=0.12∗10=1.2 W; symbol for ∗ is & # 8 7 2 7 ;

### Calculating length

We have a 1m ruler and we divide it into 1,000,000 divisions. How long are those divisions?

length of division = 1m/1,000,000= 1 μm ; the code for μ is & # 9 5 6 ;

### Calculating Area

Area = π⋅r2 of a circle; the code for π  is & # 9 2 8 ;

### Calculating circumference

Circumference = 2π⋅r of circle