Triangles and Squares.

Deviating momentarily from my Colours series for a bit of a maths/nerdy post here.

A while ago, I was wondering about how to calculate how many squares of any size were in a square grid. Take, for example, this one:



That's obviously a 4 by 4 square grid. So you would need to find all the 1x1 squares (which is obviously 16), but also all the 2x2, 3x3, up to 4x4 (just 1).
I actually figured this out when I was sitting on the toilet, as you do. Because of the bathroom tiles. (We have square white tiles in the bathroom.) I figured out that I could do it by working from the outside in - first counting all the spaces, then the grid points, and repeating that until I got into the middle - each time I had done it, disregarding the outer ring.
So, for the grid above, there are 16 spaces; then 9 intersecting grid points; then 4 spaces, removing the outside ring; then 1 grid point, again removing the next ring.
I quickly realised that this pattern (16 + 9 + 4 + 1 = 30) was a sum of squares, up to the length/width of the square itself.
I knew that that could be expressed as x^2 + (x-1)^2 + (x-2)^2 + ........ + 2^2 + 1^2. But it was tricky trying to get it down any further. Thankfully, after a bit of hunting around, I was able to find this formula: (x(x+1)(2x+1))/6, which works perfectly. And, it means you don't even need to see the grid - you just need to know that it's a perfect square, and what the length of it is in the unit squares.

Then, however, I wanted to move on to triangle "grids", as such. Which proved to be a lot harder. Take, for instance, this one:

I was able to add up the numbers in a similar way as I did the squares - except, it had three different things to count; top-pointing, bottom-pointing and then the intersecting points (so this one, for example, would be 16 + 6 + 3 + 1 = 26) - but the numbers weren't in a similar series to before. The square seemed to always be there at the beginning (the triangle above has a side length of four, for example; you only count the top-pointing triangles), but then it was a different series that I didn't quite recognise. It was, however, consistent. So I looked at it a bit closer. This is the series: 0, 1, 3, 6, 10, 15, 21.... I shortly realised that the difference was simply increasing by one each time. However, what each next triangle had wasn't just the next number in the series - it was all the previous ones as well. So the fourth triangle wasn't just the fourth square (16) plus the fourth in this series (6), but the first four in it (0, 1, 3, 6). Which was a bit harder to try and figure out a way to write down. So I tried adding them together, to see if that created a new series. I got: 0, 1, 4, 10, 20, 35, 56.... Which was better. So now, I could express the sum of the triangles as: x^2 + x(N), N being that series. But I wanted to do better than that. So I looked the series up, and found that they were tetrahedral numbers. And could be generated with the formula: (n(n+1)(n+2))/6, which is quite similar to the previous one I had for the sum of squares. However, this generator started the series at 1. I needed to start at 0. Which meant that n = x - 1, which I substituted in to the equation:
= ((x-1)((x-1)+1)((x-1)+2))/6
= ((x-1)(x)(x+1))/6
= ((x^2-x)(x+1))/6
= (x^3 + x^2 - x^2 - x)/6
= (x^3 - x)/6
...Which actually works. And is a surprisingly simple solution.

So, the equation for the calculation of all the squares of any size in a square grid with the number of unit squares along the side being given as x is equal to:
(x(x + 1)(2x + 1)) / 6

And the equation for the calculation of all the triangles of any size in a triangle "grid" (if you have a better word, let me know) with the number of unit triangles along the side being given as x is equal to:
x^2 + ((x^3 - x)/6)

I'm also planning to figure out how to do this equation with sections of the square or triangle missing - so they are rectangles or trapeziums. And you are given the ratio of one side to the other. Seems like there should be a way to do that.

Red.

You can be red in the face; caught red-handed; red hot; read the red letters; get a red alert; but hopefully, you're not in the red.

Red is stop. That's become a deeply ingrained association with the colour, and relates in to a lot of the other meanings; particularly those of danger, or warning. The other two main associations with red are blood, and fire; the latter from which also stems the ideas of passion and emotion.

Common connotations with red are danger, alert, warning, stop, fire, heat, spice, passion, emotion, love, lust, shame, blood, death, sin, and often rose, or apple. The specific connotations may also depend upon the shade or tone; darker shades may be more reminiscent of blood and death, while lighter ones may be leaning towards passion and emotion.

To me, red more represents danger than passion. A warning; or something hot, like a fire. Flowers and food also come to mind - my mum is quite an avid rose grower; and red is quite common in tomato, capsicum, chilli, and some spices as well. Personally, it's not really a colour that I like that much, but that's also because I prefer cool colours. It is a lot better than a grey or black, though.

The song for today is Red Hot, by Jimmy Barnes. Bit of an oldie, but pretty good.

Black.

You can be back in black; get a black eye; be black as night; get a black licence; buy something off the black market; have a black mark against your name; be cursed with the black spot; and nowadays, we want to paint it black.

Black can be thought of in a couple of different ways; as the presence of all colour, or the absence of all light. The former brings to mind thoughts of being very busy, cluttered, and noisy. The latter is dark, shady and suspect. The latter seems to be more where most of the thoughts of black reside. Though, interestingly, it seems that a good few opinions on black are drawn from people's opinions of night - and because the two seem to be so synonymous, the one becomes the other.

Generally, the connotations with black are words like deep, dark, bad, evil, night, sin, despair, pain, suffering, torture, and sometimes violence. It also relates in to extreme difficulty or challenge (like a black diamond ski run). More recently, it has also been related in to cool, quiet, smooth, calm, and other similar words - these, I believe, have been derived more from people's perceptions of night than black in itself. Though "quiet" could be in relation to ninjas. Some also relate it to power, or strength.

For me, black does bring to mind the worst of times. The darkest of places. The hardest of obstacles and paths to tread. But, in the same way, it reminds me that if this is the worst, things can only get better; otherwise they wouldn't be the worst. If you're in the blackest night, even the smallest light can seem like the brightest star.

For each colour, I'm also going to have a song to go with it - taking a bit of a leaf from ol' Mozart's book. Today, it's "All Black" by Good Charlotte.

True Colours.

There is meaning in everything. And because of that, I think it can be helpful sometimes to go back to simpler things, and look at the meaning in them; because they still have so much meaning, and such deep meaning, but it can be a bit less convoluted and complex. Ish. Sometimes.

Colours are a great example. We see them everywhere. We have so many names for them, that have become so long and obscure - like magenta, azuline and caesious. They pop up quite often in various puns, metaphors and similes. We can choose so much based on colour - clothes, food, furniture, cars, a house - for some, even people. (Though that's typically a little different. Unless it's hair colour.) So I thought it might be an idea to have a look at a few of these colours.

But which ones to choose? Well, black and white are the two extremes - probably good to put in. Then the primary colours; but which ones? RGB (red, green, blue), as according to light and TV/PC systems? Or red, yellow and blue, like we get taught at school, because that's what works with paints and crayons? Well, let's put in all four. So I'll be doing white, yellow, green, blue, red, and black. Maybe that order, maybe reverse. We'll see.

So I'm going to do one post on each of those colours. But then, there are a lot of other colours that do have meaning - old and grey; royal gold and purple; fruity orange; girly pink; sea turquoise. But I suppose for me, most other colours have meaning that is more shallow when compared to the ones that I'll be focussing on. For some people, certainly, that won't be the case - but in a more general sense, that seems to be how it is; so I am restricting myself to these six.

Stay tuned for the colourful posts to come.

Aiyæthron.

These are my notes and translations thus far for this language that I'm making, which I've called Aiyæthron. Pronunciation of vowels can be found just below. Hopefully, should be reasonably self-explanatory. That one, however, is pronounced Are-ear-yair-throrn. (Short r sound at the end; don't emphasise. Emplicit in vowel sounds in this language, but not emphasised unless written.)

* Italics are currently ponderings as to possible solutions to difficulties encountered.

** [Comments in square brackets are added in to aid in reader understanding of my logic.]

Open/vowel sounds

A - ar - "are"

E - er - "err"

I - ir - "ear"

O - or - "oar/ore"

U - ur - no Eng. equiv. (N.E.E.) [oor]

Æ - ær - "air/e'er" [I do realise this is not the typical pronunciation of this letter]

Apostrophe ( ' ) cuts short; write h afterwards for a h sound instead of an r sound? [ah, eh, ih, etc.]

Closed/consonant sounds

Combiners - L R W Y S Z

DR => JR [compare dive/jive to drive/jrive)

TR => CHR [compare tane/chain to train/chrain]

K/Q => C/CW [All K sounds replaceable by C, as well as hard Q sound {e.g. Qatar, Iraq}; QU sound replaceable by CW - almost is for Qwerty]

X => Z/CS/GZ [Initial X {e.g. Xavier, Xanadu} replaceable by Z, intermediate/ending X replaceable mostly by CS {e.g. Hex, Explain} and in some cases by GZ {i.e. Example}]

BL BR BS BW BY BZ

CL CR CS CW CY CZ

CHL CHR CHW CHY

DS DW DY DZ

FL FR FS FW FY FZ

GL GR GS GW GY GZ

JR JW

MS MW MY MZ

NS NW NY NZ

PL PR PS PW PY PZ

RS RW RY RZ

TL TS TW TY TZ

THL THR THS THW THY THZ

SB SC SCH SD SF SG SL SM SN SP SR ST STH SV SW SY

SHL SHR SHW SHY

VR VS VW VY VZ

WY

ZW ZY

ZHL ZHR ZHW ZHY

Alphabet

Æ A B C CH D E F G H I J L M N O P R S SH T TH U V W Y Z ZH

Words

ai - life/wind

yæ - love

thron -> thro-on

                l      l

     (sounds) + (perspective) = words

Aiyæthron - words of life and love

urmdwur -> urm-dwur

                     l      l

       (plant life) + (guardian) -> guardian of plant life -> earth

oayio -> encircler of life -> air

oyo - circle

leæl -> le-æl

            l    l

 (animal) + (provider) -> provider for animals -> water

zhrng -> zhr-rng

               l     l

   (destroy) + (heat) -> heat that destroys -> fire

Comments welcome.

On Creating A Language.

I watched this video the other day: http://www.youtube.com/watch?v=Uq-FOOQ1TpE
And one of the things he was saying in it towards the end was, for a day, to not learn anything. Just stop learning - and start thinking. Go away and think about what you really enjoy and our passionate about, through your own unique perspective, and create something new.

And that inspired me to make a language.

It's actually something I've had in my mind to do for a few times before, but I've never gotten around to it. But now I've started. And it's....pretty tricky, particularly because I get so finicky about it.

Because you could just put English through a cipher or something, and say that you've come up with a new language. But it's not really a new language. I wanted something where each word had real meaning in the way that it was constructed; that made sense and sounded right to me.

So, first I had to decide what sort of language I was going to make. Because there are different types, you see; for the different senses, really. You can have a written language; a spoken or auditory language; a signed language; or you could even create a language based around smell, taste or touch, though that would be more difficult. (I'd still call Braille a written language myself, even though it is perceived through touch.)

I decided that I wanted a language that was primarily auditory - I'll probably write it down at some point and create an alphabet for it, but I wanted the words to really be formed based on their sound.

I also wanted to make a language that was based around small building blocks, as it were; not so much an alphabet as a library of sounds that could be combined. So first I had to make the library. That's harder than it sounds.

I worked out all the open (vowel) sounds - and then the closed (consonant) sounds. In doing that, I also sorted out a few letters I wouldn't be using, a few I added in, and a couple of letter combinations that wouldn't occur either.

I'd had a name for the language already in mind, almost from when I had the idea to create it. But then I had to actually break down the word into its parts, and figure out what each meant, and how it fat in. (Because fat should be the past tense of fit. It just works.)

Then I wanted to figure out the words for earth, air, fire and water. Thought it might be a good way to start. But the way I decided I wanted the language to work, is that a word, in a sense, describes something - the definition is implicit in the word. My word for fire, for example, was made up of two smaller parts that meant "destroy" and "heat". Put together, it meant "the heat that destroys": fire. Still need to work out the "the"s, "that"s, "and"s, "is"s and so on, but you get the idea. The point of that is also so that if something sees something that they've never seen before - and as such, don't have a word for - they can create a word for it straightaway by using these shorter describing bits.

That, of course, makes interpretation quite difficult. Because while I may call fire "the heat that destroys", someone else may well call it "illuminator" or "man's greatest invention". As such, there are feasibly millions of different words for the one thing.
As such, I'll probably come up with a few standardised words as well, to aid in communication.
But essentially, it means that rather than writing a whole new dictionary, I'm just making those short describers that people then put together to form their own words, their own language, as such. I guess that's the idea.

It's going to take me a long time - and may well never be finished - but I'm quite enjoying the challenge.

I'll put up another post shortly with all the notes I've got on the language insofar.