An Etymological Introduction to Trigonometry

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Figure 1:  The mathematician, George Boole (1815-1864), was self-taught and fluent in Latin, Ancient-Greek, and Hebrew by the age of 12.  I will be 30 in less than a month, and I am not even close to being fluent in any of these languages.  However, I still cultivate an interest in these languages in some measure of poor imitation of the great man.  It is certainly a great irony that subjects thousands of years old, such as Ancient-Greek and Latin, can make something so modern, such as Computer Programming, so much easier.  If you have an interest in Science Fiction, you will notice that even in the far-flung future, scientists will name their bionic monsters after letters of the Ancient-Greek alphabet.  The letter ‘Sigma,’ or Σ seems to be a favourite of Science-Fiction writers.  In Ratchet and Clank: A Crack in Time, the Robot Junior Caretaker of the Great Clock is called Sigma 0426A.

Figure 2:  Ecstasy tablets, very often, have the Ancient-Greek Majorscule, Sigma, stamped into them.  So it is not only Computer Programming that a knowledge of Ancient-Greek will make easier: it will also give you a head start in Pharmacy!

I shall be studying QQI Level-V Videogame development in September.  One of the modules of which this course comprises is called:

Mathematics for Programming

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A huge part of this Maths module is Trigonometry.  This is why I wish to develop an implicit knowledge of the fundamentals of Trigonometry, now, prior to beginning the module formally, in September.

The purpose of this article is to take a look at the etymological meaning of some of the key terms pertaining to Trigonometry.

‘Trigonometry’

The term, ‘trigonometry’ is derived from four root words:

1. ‘trí’ This is the Ancient-Greek Cardinal Numeral, 3.
2. ‘tó gónu’ This is an Ancient-Greek third-declension neuter noun, which means ‘the knee;’ ‘the corner;’ ‘the vertex.’
3. ‘tó métron.’ This is an Ancient-Greek second-declension neuter noun, which means ‘the measurement.’
4. ‘-y.’ This is a noun-making suffix.  It comes from the Latin substantive-adjective 2^nd-declension neuter plural suffix ‘ [i]-a.’

Figure 3:  Etymologically, ‘Trigonometry’ is the study of the measurement of three-cornered polygons, or triangles.

The four root words, or etymons, listed above, when considered together, give us an etymological definition of ‘Trigonometry:’

The study of the measurement of three-cornered polygons.

The study of the measurement of polygons with three vertices.

The study of the measurement of triangles.

Now that we have the term ‘Trigonometry’ broken down, etymologically, let us now consider the Trigonometric term, ‘equilateral.’

‘Equilateral’

Figure 4:  An equilateral triangle.  An equilateral triangle has sides of equal length, and angles of equal magnitude.  Each of an equilateral’s interior angles is equal to 60º in magnitude.

The term, ‘equilateral,’ can be broken down, etymologically, into three root words:

1. ‘æqua, æquus, æquum.’ This is a 1^st-and-2^nd-declension Latin adjective that means ‘equal.’
2. ‘latus, lateris.’ This is a 3^rd-declension neuter Latin noun that means ‘side.’
3. ‘-ālis, -āle.’ This is a 3^rd-declension Latin adjectival suffix.

The three root words, or etymons, listed above, when considered together, give us an etymological definition of ‘equilateral:’

 

of [triangles] that possess equal sides[1]

Now that we have the term ‘equilateral’ broken down, etymologically, let us now consider the Trigonometric term, ‘isosceles.’

‘Isosceles’

Figure 5:  An isosceles triangle.  An isosceles triangle has 2 sides equal in length, and two angles equal in magnitude.

Figure 6:  An isosceles triangle has two ‘legs’ or sides equal in length.

The term, ‘isosceles,’ can be broken down, etymologically, into two root words:

1. ísos.  This is an Ancient-Greek adjective that means ‘equal,’ ‘the same,’ ‘proportionate.’
2. tό skéllos.   This is an Ancient-Greek third-declension neuter noun that means ‘leg.’

It is funny how, in Ancient-Greek, the sides of triangles are called ‘legs’ and the corners of triangles are called ‘knees!’

The two root words, or etymons, listed above, when considered together, give us an etymological definition of ‘isosceles:’

A triangle [that possesses two sides] that are equal in length.

A triangle [that possesses two sides] that are the same in measurement.

A triangle [that possesses two sides] that are proportionate.

Now that we have the term ‘isosceles’ broken down, etymologically, let us now consider the Trigonometric term, ‘scalene.’

‘Scalene.’

Figure 7:  A Scalene Triangle.  As we can see from the above diagram, a scalene triangle is one which possesses 3 sides, all of unequal length; and 3 interior angles, all of unequal magnitude.

Figure 8:  The Trigonometric term, ‘scalene’ is derived from the Ancient-Greek adjective, ‘skălēnos,’ which means ‘unequal.’

The term, ‘scalene,’ can be broken down, etymologically, into the root word:

1. ‘skălēnḗ, skălēnos, skălēnón’ This is a 1^st-and-2^nd-declension Ancient-Greek adjective that means: ‘uneven,’ ‘unequal.’

 

When we consider the root word, or etymon, listed above, then we can come to the following etymological definition of ‘scalene:’

[of a triangle whose sides are] of unequal [length.][2]

 

 

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[1]  Et sequitur: equal angles.  It follows, according to mathematical logic, that if a triangle’s sides be all equal in length that its interior angles will, likewise, be all equal in magnitude.

[2]  Et Sequitur: and whose interior angles are unequal in magnitude.  It follows, according to mathematical logic, that if a triangle’s sides be all unequal in length that its interior angles will, likewise, be all unequal in magnitude.

The Logical permutations of an Inclusive-Or Gate Realised in JavaScript:

Figure 1: The JavaScript Logo. I drew this with pencils.

Figure 2:  The schematic symbol for an Inclusive-Or Gate.

The logical permutations of an Inclusive Or Gate are as follows:

0 ∨ 0 = 0

0 ∨ 1 = 1

1 ∨ 0 = 1

1 ∨ 1 = 1

The above can be read, in English, as follows:

Zero disjunction zero equals zero.

Zero disjunction one equals one.

One disjunction zero equals one.

One disjunction one equals one.

Alternatively:

Zero OR zero equals zero.

Zero OR one equals one.

One OR zero equals one.

One OR one equals one.

We can realise the above Boolean Logic in Digital circuitry, as follows:

Figure 3:  An Inclusive-Or Gate.  This represents the Boolean Equation, 0 ∨ 0 = 0

Figure 4:  This Inclusive-Or gate represents the Boolean Equation, 0 ∨ 1 = 1

Figure 5:  This Inclusive-Or gate represents the Boolean Equation, 1 ∨ 0 = 1

 

Figure 6:  This Inclusive-Or gate represents the Boolean Equation, 1 ∨ 1 = 1

It is possible to translate Boolean Equations into Conventional-Arithmetic Equations, and to obtain the same logical result:

x ∨ y = x + y – ( ( x ) * ( y ) )

0 ∨ 0 = 0 + 0 – ( ( 0 ) * ( 0 ) )  = 0

0 ∨ 1 = 0 + 1 – ( ( 0 ) * ( 1 ) )  = 1

1 ∨ 0 = 1 + 0 – ( ( 1 ) * ( 0 ) )  = 1

1 ∨ 1 = 1 + 1 – ( ( 1 ) * ( 1 ) )  = 1

It is possible to use the above logical translations so as to create buttons in JavaScript that calculate the logical permutations of an Inclusive-Or gate.

Figure 7:  In the above HTML-5 file, we declare four functions.  Each function describes a logical permutation of the Inclusive-Or Gate.

Figure 8:  What the JavaScript and HTML-5 code outputs.  We have created 4 buttons.  Each button calculates a logical permutation of the Inclusive-Or gate and displays the result in a dialog box.

Figure 9:  We are about to click the button that calculates the Boolean Equation, 1 ∨ 1 =

Figure 10:  The logical result of the Boolean Equation, 1 ∨ 1 = , i.e. 1, is printed to screen in a dialog box.

Figure 11:  A screenshot of a web-application that I programmed, using the code depicted above.  Click the following link: Inclusive-Or Web-application , so as to try this web-application out for yourself!

Must Do is a Great Master.

The title of this piece comes from a Monaghan-ism that has a parallel in non-localised English.

As part of our Networking-and-Systems Apprenticeship, we have to draw Network Topologies.  I am watching a Microsoft MVA video series on how to do this:

Video 1:  A Microsoft Video Playlist on Networking Fundamentals.

The chaps at Microsoft recommend some of their proprietary software for the task: Microsoft Visio.

I looked up Microsoft Visio on their online store, and, to my horror, it cost in excess of €700.  It amazes me that triple-A videogames can employ hundreds of developers that can work years of man-hours between them, and, yet, they can retail their software at €60-€70.  How come non-gaming software can sometimes retail at the €1000+ mark?

So, I am not spending money – that I do not have – on this.  Eventually, I would like to set up a development company using the Microsoft Bizspark program, and, I think, Microsoft will give me all of their software for free, upon the proviso that I develop for their platform.

So I looked for a coding solution.

I took a low-quality raster screenshot of the server icon featured in their video:

Figure 1: A low quality raster-screenshot of a server icon.

I then began to manually describe this Server Icon in SVG[1] code:

Figure 2:  A manually programmed svg file that I produced.

Below is a capture of the high-quality vector image that the code produced:

Figure 3:  A capture of what the simple svg file, above, produces.

Sometimes, lacking the proper resources necessary to – ordinarily – complete a task, forces one to seek more creative solutions.

As in the adage:

Necessity is the Mother of Invention

Indeed, it is necessity – and not plenty – that it the mother of invention and innovation.

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[1] Scalable Vector Graphics.

Deliver me from Arkham.

I am beginning to read the fiction of H.P. Lovecraft.  I was introduced to H.P. Lovecraft by way of Southpark.  There is a trilogy of episodes in which Cartman teams up with Cthulhu to wreak havoc upon the planet.  It is up to Kenny to stop him, which he eventually does in the concluding episode.

Figure 1:  South Park has been a civilizing influence upon me!

In H.P. Lovecraft’s Cthulhu-Mythos universe, the fictional New-England town of Arkham is oft-referred to.

It is from the Ouevre of H.P. Lovecraft, therefore, that we derive the name of the frontier psychiatrist, Amadeus Arkham, and his Asylum, in the D.C. Universe.

This brings us nicely on to the 2009 gaming masterpiece, Batman: Arkham Asylum.  I have, also, read the 1989 comic book, Arkham Asylum: A Serious House on Serious Earth, and the 2009 Rocksteady Game actually follows this graphic novel extremely closely.  In both the game and the comic, the Lunatics take over the Asylum; a backstory of Amadeus Arkham is given; and Batman must wrestle with questions regarding his own sanity, or lack thereof.

Arkham Asylum (2009) is a marvellously gothic and creepy game, but, I believe that the most gothic and creepy aspect of the game is a visit to Death Row in the Arkham Penitentiary.

Figure 2:  A visit to death row.  A gaming highlight. I drew this with pencils.

Wandering through Death Row, the player finds Latin inscriptions upon the wall.  One of these Latin inscriptions is:

Līberāte mē ex īnfernīs.

, which is Latin for:

Deliver me[, ye gods] from Hell.

The above inscription sends a chill down the player’s spine.

Another Latin inscription that the player finds upon the wall is:

Mors certa, hōra incerta.

, which is Latin for:

Death is certain, the hour unceratain.

I love this quote!  How ironic that this should be found inscribed upon a death-row wall, where the hour of death is, pretty-much, very certain indeed!  What terrific gallows’ humor!

And the game does not pander to the player, at all.  The above phrase is never translated.  The player either has Latin sufficient to get the joke, or he does not!

Figure 3:  The Family-Guy logo.  I drew this with pencils.

I write, quite a bit, about Family Guy on this blog.  I listen to the DVD commentaries, when I view this show.  Comedy Writers’ call obscure jokes:

one-per-cent-ers.

Only about one per cent of the audience will get the joke, but this one per cent will feel really privileged, and special, as though the joke was written with him in mind, so much so that he will tell all his friends about the show, thus generating free advertising.

This is precisely how I feel about the:

Mors certa, hōra incerta.

joke!

Andrew Ryan is Basically Macbeth.

Figure 1:  An edition of William Shakespeare’s The Tragedy of Macbeth identical to the one that I studied from so as to sit the Leaving Certificate 2004 English Examination.  I have read this play a number of times since then.

Andrew Ryan is very much the sympathetic villain. He is like Macbeth. We cannot help but like the guy and partially agree with some of his ideas and modus operandi. Ryan is anti-war; anti-undue-state-interference in the lives of a state’s citizens; anti-conscription; and anti-nuclear weapons. These ideas are growing in popularity in the Avant-Garde Internet Thought tribes of the 21st century.

Ryan Pioneers free non-polluting sustainable energy – the Hephaestus Volcanic Energy Station – although he was a blackguard for charging money for the use of free energy.

Ryan submits willingly to his demise – his being bashed on the noggin with a golf club – without the slightest resistance. This also earns our sympathy. In the audiotapes we hear Ryan’s anguish over the medical-and-physical abuse of “the little sisters.” He expresses anguish, and yet continues the practice.

Ryan is a “tyrant” just like Macbeth is described as being.¹

Ryan is just evil enough to be a villain – and not an anti-hero, say – and just good and noble enough so as to earn our respect and sympathy.  Ryan is very like the eponymous protagonist in Shakespeare’s The Tragedy of Macbeth.

Figure 2: During the Bioshock Game, Andrew Ryan will ring you, the player, Jack, from time to time, on a radio device.  This avatar of his pops up on screen whensoever that occurs.  This avatar also pops up on screen whenever you, the player, Jack, listen to one of Andrew Ryan’s Voxophones.  I drew the above with pencils.

Figure 3: The Bioshock Games Feature loads of references to Greek Mythology.  “Neptune’s Bounty” is where Fontaine Fisheries is located.  The “Hephaestus” plant, the volcano-heating-and-energy-generation system is named for ‘Hephaestus’ whose Roman equivalent is ‘Vulcan’ whence we derive the English word, ‘Volcano.’

Figure 4:  Andrew Ryan agonizes over the medical-and-physical abuse of “the little sisters,” i.e. the taking of orphan girls, and the introducing of parasitic sea slugs into their stomachs so as to produce the genetic-splicing material known in the game as “ADAM;” but he continues on regardless.  He is like Macbeth in this regard, who agonises over the regicide, the killing, of the just King Duncan, but who does so anyway.  Andrew Ryan is like Cornelius, who, in Hamlet: Prince of Denmark expresses regret that he usurped the Kingship of Hamlet Senior, but who continues on with the usurpation regardless.

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¹ Andrew Ryan is a tyrant who runs Rapture in the same ruthless and murderous fashion that Macbeth ran early Medieval Scotland.

Notes on Ruby

I have downloaded RPG Maker XP.  I am thinking of constructing a Computer Role-Playing Game of Joseph Sheridan Le Fanu’s Carmilla.  I can just imagine the creepy village of Carnstein rendered in glorious 16-bit-esque top-down graphics.  If anyone wishes to collaborate with me on this, please contact me: mathsandcomedy@outlook.com

Below are a few notes on Ruby, the programming language that RPG Maker XP runs on.

Figure 1:  I drew this with a mechanical pencil and ruler.  I then coloured it in with an oil pastel.  Thereafter, I scraped off the excess pastel with a craft knife.  This has given this drawing of a gem a wonderful illusion of translucency, and light refracting within the gem.

Notes on Object-Oriented Programming: (I researched these notes from the Oracle,– Java – website.)

• We must conceptualise States and Behaviours.
• A State is what an Object is.
• A Behaviour is what an Object does.
• A State can be a Variable, such as:

x

• A Behaviour can be a Function, such as:

f(x) = x + 2

In Programming, a State, or Variable is analogous to a Field.

In Programming, a Behaviour, or Function is analogous to a Method.

Ruby is an object-oriented programming language.

Notes on Idiom.

• In Ruby, we do not use tabs to indent, we use spaces.  The number of spaces that we use to indent a line of code is 2.

Mister Bubbles

 

Figure 1: Little Sister and Big Daddy.  This could be a postcard from Rapture.

In this game, the player must save the Little Sisters. It is based on the Novel by Ayn Rand -that I am yet to read – Atlas Shrugged.

An in-game ally is named “Atlas.”

The game’s adversary is Andrew Ryan, which is an anagram of “We R Ayn Rand.”

The underwater city, Rapture, contains numerous references to Graeco-Roman Mythology: Minerva’s Den; Neptune’s Bounty, etc.

 

Figure 2: It is not surprising that the Graeco-Roman Lord of The Sea should be referenced in a game that deals with an underwater city.

Figure 3: Neptune is also the deity to whom the residents of Bikini Bottom, Spongebob Squarepants and company, pray to.

I have an essay floating about in my head that will treat of Rapture’s economic worldview.  I just need to write it down.

Figure 4: One of the game’s Easter Eggs is the tombstones of Economists, Adam Smith and John Maynard Keynes, in the tea garden.  By this, the game seems to suggest that current economic paradigms have run their course.  What will replace them though?

 

A Rush of Unexpected Reward.

A Rush of Unexpected Reward

The Above Link is The Microsoft Word version.

On a neurological level, there is scarcely anything so pleasurable as the rush of Dopamine that is obtained by the attaining of a doubted goal.
Imagine piecing together a jigsaw puzzle. Regardless of how many pieces it may have, the outcome is never in doubt. The pieces will come together to form an integral whole: it is only a matter of time. When your shaking hand places that last piece into the 5,000-piece puzzle, the chemicals or neurotransmitters that rush around in your brain make you feel pleasant, but not manic. This is the meagre thrill of attaining an expected outcome.
When betting on a horse, ceteris paribus¹ you only have one chance out of the number of horses racing. The Grand National took place at Aintree, Liverpool, England, not so long ago. The Grand National is generally regarded as one of the most open horse races on the planet. There are so many obstacles and variables, that, practically any horse could win it.
If you placed a bet on the grand National today, you would have had a one in forty chance of seeing an accrual of winnings. That is a 2.5% chance.
To restate: the chance of completing a 5,000-piece jigsaw = 100%; the chance of winning a bet on the Grand National is 2.5%
There must be a part of the brain that subconsciously calculates odds, as it has now been proven that the thrill that one gets from an anticipated reward is inversely proportional to how statistically likely that anticipated reward is to occur.
When I fixed my U.S.B. charger, at the outset I gave myself, subconsciously, odds of about 1 in 10,000 of success. I had never done the like before. It was in a sealed unit and not meant to be fixed. The insulation clinging to the two copper wires was very difficult to remove. As the task progressed, my subconscious odds plummeted: I cut straight through the insulation and right through the stranded copper wire a few times.
Whenever I plugged the repaired U.S.B. cable in and found it to work I felt an overwhelming euphoria inundating my mind! I had bravely defied those Chinese industrialists who sealed this unit in plastic so that it may never be mended, if broken. I had learned a new skill. I had unexpectedly succeeded at something. The brain rewards you – with sweet, sweet dopamine – for learning new skills. Likewise it rewards you for unexpected successes. Indeed, a company called Zondle is attempting to introduce the element of unexpected reward into its teaching of youngsters.
In my opinion, that is why many people are addicted to gambling. They are not addicted to the money that they might sometimes win – as rational thinking adults they know that on a long enough timeline the house always wins – they are addicted to the dopamine rush that the few-and-far-between wins on horses and soccer matches produces in their brains.
If one were to think of Casinos and Bookmakers’ Shops as taverns, but instead of purchasing Alcohol, the patrons purchase the highly addictive neuro-transmitter Dopamine, then this would be a good way of conceptualising the psychological disorder of Addiction to Gambling.
A rush of dopamine is also what occurs when one defeats an extremely difficult video-game level. A couple of weeks ago, I finally beat the ‘vigilante’ level in Grand Theft Auto: San Andreas. I have been trying to beat it for a number of months now. On several occasions I deemed it impossible. On the final part of the level, you are pitted against three automobiles, the passengers of whom can shoot you with shotguns from out their windows. It only takes three or four shots, discharged in a couple of seconds to finish you. I discerned this an impossible obstacle.
When I finally -narrowly!- overcame this impossible obstacle I felt utterly elated. I felt a warm-and-toasty dopamine buzz for hours afterwards. Everything felt right in the world, and I felt that I could accomplish anything if only I showed the same determination and tenacity to real-life problems as I had shown to the problem of skooling the digital bruthas of San Andreas in the art of pitiless vigilante justice.
I played the ‘vigilante’ level hundreds of times in vain. I managed to succeed in it once. That means that – on the whole – my chance of succeeding at the level on any one attempt was about 0.1%, a one-in-one-thousand chance. Finally succeeding in something after the thousandth attempt constitutes the thrill of unexpected reward, and the neurotransmitter glands in your brain reward you handsomely for it.
What surprised me – initially – is that I seem to acquire the same dopamine buzz from the solving of Mathematical Problems. Often I attempt – what is to me at least – to solve a very complex problem and fail miserably. What usually happens next is that I let the unresolved mathematical issue ferment in the back of my mind for a day or two; attempt the problem again; and invariably solve it. I feel marvelous, and award myself an ojis big tick. This is the thrill of unexpected reward. Something that had once utterly bested me I had come back to defeat.
It is like the vigilante mission in San Andreas.
The benefits of employing Mathematics to obtain my dopamine fix instead of Computer Games is manifold:
I am learning a sale-able skill (such as being able to count in bases other than decimal) rather than an un-sale-able skill (such as being able to finish Crash Bandicoot 1 without losing any lives – exceedingly difficult! – so as to gain 100% completion).
I can generally spend longer studying Mathematics, as reading and writing is not as tiresome on the eyesight and mind as controlling an image progressing at 60 frames per second.
After doing Maths I feel a sense of accomplishment. After spending an immoderate amount of time playing computer games I feel a sense of regret at not having employed my time better.
Whereas, not so long ago, I would arise, play videogames and go to bed, and that would be the story of my day, I now use videogames as a reward mechanism. Imagine a DualshockTM controller suspended in front of my face instead of a carrot² . I now make myself earn videogames. I learnt this trick from a program on T.V. as a child. In some far eastern country whose name escapes me, it is the custom for young boys to be brought up by Buddhist Monks. They have the option of staying on at the monastery when they graduate should they so wish. A monk was on camera explaining that treats such as videogames had to be earnt by means of performing productive tasks such as schoolwork etc. The Camera shows a young boy playing Bomberman³ , or a game somewhat similar. The monk explained that this was a good way of instilling discipline in the male children.
All videogames are the visualisation of Mathematical problems. All computers are binary machines that operate on Boolean Logic. A computer shows you an arithmetic problem, but it does not show you the numbers, it shows you graphics that correspond to the numbers. You then solve the arithmetic problem by inputting a specific code. However, you do not type in numbers, you press buttons on your game controller that correspond to the arithmetic solution to the above arithmetic problem.
But, you may say, I can complete this level in a number of different ways, whereas a sum only has one possible result.
Not necessarily. In some mathematical calculations there is a range of acceptable answers.
You are doing sums, Jim, but not as you know them.
If you enjoy computer games, then you enjoy Maths, but you may not have just figured that out yet!

[1] All things being equal.  From the Latin adjective ‘cēterus, cētera, cēterum’ meaning ‘other,’ and the Latin Adjective ‘pār, paris,’ meaning equal.  In the phrase ‘cēterīs paribus’ the Neuter plural form of ‘cēterum’ is being used substantively, i.e. ‘other thing.’  ‘Cēterīs paribus,’ is a Latin Ablative (adverbial) phrase meaning: ‘with all things [being] equal.’

[2] Dualshock^TM is the brand of Video-game controller used by Sony Playstation.

[3] A video-game franchise that commenced in 1983.  The object of this game is to get an amateur demolitionist, a bomber man if you will, to blow up bricks and enemies with bombs.

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Who is The Real Villain

Yesterday Evening, I purchased Yoshi’s New Island for the 3DS. I was playing it, and I could not believe what one of the ‘enemies’ looked like. He was a singing, walking flower. Imagine this: a flower is out for a stroll; singing a merry tune; minding his own business, when, all of a sudden, he is metabolised by a fearsome dinosaur in the form of Yoshi. Yoshi need not eat him: if Yoshi jumps – or, worse still, stomps – upon this gentle bloom, then he disintegrates into a cloud of petals.

Take a look at the accompanying illustration.  This is a sixteen-bit rendering. This is how this enemy appeared on the Super Nintendo Entertainment System.  Have you ever seen a less malevolent-looking video-game enemy in all your life?

I drew this ‘plant enemy’ using graph paper; a Pilot V5 Hi-Tecpoint 0.5 Pure Liquid Ink Black Pen; and some Staedtler Triplus Fineliners.