The simplest unit of spatial thought ... is the Right Angle

[pyxxo]

When we design, create, build or arrange objects, the most basic, fundamental shape we use is the right angle and its progeny: the rectangle, square, cube and cuboid.

 

Here's the modern city of Johannesburg, beside the city of Kahun, near Giza in Egypt, built 5,000 years ago:

 

 

Both built on right angle grids.

 

Here's some game boards: chess, tic-tac-toe, sudoku, a crossword, scrabble:

 

 

all played on square grids.

 

The screen you're viewing this post on, all the windows it contains, the menus, the icon, character, button and picture spaces, and the pixels themselves ... are, from a visual standpoint, all designed with the right angle as the most rudimentary concept.

 

Here's various sports pitches: basketball, football, badminton, snooker, volleyball:

 

 

All rectangles. If you're not commuting right now, chances are you're reading this inside a room... a room made from right angle corners, in a building built with right angle corners ... all made from right angle bricks.

 

Here's an exercise for you do. Take a piece of paper, a pen, and multiply 34 by 27. *Don't* use a calculator. Do it manually. Don't read any further until you have the answer.

 

 

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When you wrote out the calculation, you organized the numbers into a right angle grid. You put each number (apart from the carried numbers) into it's own imaginary, rectangular box. In school, you learnt this on squared paper. Look back at the paper now, and draw the lines between the numbers where the squared lines would have been.

 

 

Now pause for a moment and imagine how many times per day, across the world, a graph is shown:

 

 

and that graph is almost exclusively a right angle graph.

 

Why not draw the graph at 85 degrees? We could, of course, display the same information, with the same data points, on a graph at 85 degrees, and it would be perfectly valid:

 

 

But we don't. We show it at 90 degrees. Because 85 degrees feels wrong. And 90 degrees feels right.

 

The human mind sees 90 degrees as the most perfect angle, simply because at it's root, spatially, the right angle is the simplest, smallest unit of mental processing.

 

All of this is explained further in this video

 

Please watch the full video set before replying here. Thanks.

Edited March 23, 2009 by pyxxo

[Sayonara]

Don't you think that the efficiency benefits of regular geometry might be a simpler explanation?

 

Why not draw the graph at 85 degrees? We could, of course, display the same information, with the same data points, on a graph at 85 degrees, and it would be perfectly valid:

But we don't. We show it at 90 degrees. Because 85 degrees feels wrong. And 90 degrees feels right.

That's not the reason at all. The area under the line has mathematical significance. You use the most rational expression of that area because its dimensionality determines how easily that significance can be interpreted.

[CaptainPanic]

90 degrees is not a coincidence. 90 degrees is also no human invention, and has little to do with how we like it, feel about it or felt about it 5000 years ago.

90 degrees comes from 180/2 degrees... and in that light it's not so amazing. 180 degrees is a straight line, and no matter what kind of thing you invent, a straight line is always straight, and if take a point on that line, and you want to make a second line through that point and have an equal angle to both sides, then the perpendicular line is the one you will find. That's just simple maths, not psychology.

 

Incidentally, the 90 degrees is also found in nature: It is the angle between a flat surface of water and the direction of gravity. (Agreed, there is some rounding off here - the earth is not exactly round). Also, right angles occur in many places in physics (for example the Lorentz force - it wouldn't become 85 degrees if we changed our definition).

[Sisyphus]

It's also a representation of the relationship between spatial dimensions. Lines at right angles to one another have extension in entirely different dimensions, which makes them an obvious choice for any number of things, including graphs which show the relationship between different parameters. It also means that right angles have real physical significance whenever there's ever a unidirectional motion or force, like gravity: the relationship between vertical and horizontal is a right angle. Vertical being the direction of the force of gravity, and horizontal being the direction in which gravity has no effect.

[pyxxo]

Ok thanks for the input (ps. keep it coming!). Before I address your points, i'll throw a couple of curveballs.

 

Consider the upper case English alphabet: (verdana font, but any sans-serif font will have similar properties)

 

 

It contains a total of 12 curved lines, 19 diagonal lines, but 42 lines at right angles to the edges of the almost exclusively rectangular paper or screen it's displayed on.

 

The *visual appearance* of these symbols is essentially arbitrary and is devoid of significance in mathematics, geometry, physics, gravity or electromagnetic force.

 

There is no logical or rational need for the alphabet to contain 58% horizontal and vertical lines - they are purely symbols.

 

Handwriting is of course more curved and rounded, but the introduction of the printing press and mass production funnelled the english alphabet into its current, overly perpendicular form.

 

A standard capital letter 'T' is perfectly straight, true and perpendicular. Its lesser-used cousin receives a special name to denote its slanted form - the italic 'T'.

 

Script in arabic, hebrew, asian and other languages contain many more curved and diagonal lines, nevertheless, each is written horizontally or vertically across the, again almost exclusively rectangular paper or screen.

 

Consider the common staircase - which performs a simple function - the vertical transport of humans:

 

 

The horizontal portion, the tread, is horizontal for good reason - to avoid slipping. But the vertical portion, the riser, has no reason to be perfectly vertical.

The vertical riser could be slanted in either direction, like this:

 

 

The first case, the outward slant, is less safe than the standard right angle due to a smaller horizontal surface.

By that same logic, the second case, the inward slant, would be safer than the right angle version.

Yet across the world, standard staircase design remains mostly perpendicular.

 

Not because it's the best design. But because its the easiest design for humans to create and build.

 

 

90 degrees is perpendicular, orthogonal, square, perfect. Anything else, other than 180 and 270 degrees, is diagonal - imperfect.

 

Let's compare humankind's fixation with right angles to that of another animal that also spends its days creating a perfect, regular geometric shape: the bee. It's shape of choice: the hexagon:

 

 

Just like the human-made world is dominated by 90 degrees, so, the bee-created world, the honeycomb beehive, is dominated by the angle at the corner of a hexagon: 120 degrees.

 

Bees, if they could make graphs, would make them like this, with 120 degrees as the critical angle:

 

 

To a bee, 90 degrees is diagonal.

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Ok, now to your points

 

Sisyphus:

It's also a representation of the relationship between spatial dimensions

 

ie, the 3 dimensions based on 3 lines at right angles to each other, used to describe a point or points in 3 dimensional space. Which, i'm sure you're aware, can also be described perfectly well using polar coordinates.

Similarly, all of 3 dimensional space could be described using, for instance, 3 axes at 100 or 110 degrees.

 

My point, is, predominantly, we don't use those systems. We use the right angle system. That's the one we explain to kids in school. Because that's the way the human mind sees the perfectly constructed spatial template. In the video i make the statement:

 

The origin is not a blank map of the universe.

The origin is a blank map of the human mind.

 

Lines at right angles to one another have extension in entirely different dimensions

 

Again, lines at 60, 30, 78.34 degrees etc, also have extension in different dimenions.

 

 

CaptainPanic:

90 degrees is also no human invention

 

Correct. The right angle is not an invention, it's a discovery. And to label it so, we must find a right angle in nature.

And it does appear, in right angle crystals such as pyrite, flourite and halite:

 

 

In the video I make the statement:

 

Apart from right angle crystals, there are no true, natural right angles in the known universe.

 

So let's be very clear on the definition of a right angle: two straight lines, touching at 90 degrees.

 

 

Sisyphus:

the relationship between vertical and horizontal is a right angle...

...horizontal being the direction in which gravity has no effect.

 

diagonal is another direction in which gravity has no effect. Gravity only has an effect in a vertical direction. Gravity only pulls down. It doesn't pull diagonal.

 

 

Define the concept of 'horizontal' without using right angles, and using only natural concepts (apart from crystals).

 

So,

 

- "Horizontal is the direction of a tangent drawn to the surface of the earth"

 

Is not valid since such a tangent does not appear in nature. And of course, close up, the earth rarely has that perfect round curvature (due to mountains, valleys, vegetation etc). Even the seas are pulled slightly off that curvature by the moon, and the waves also negate any smoothness. Similarly,

 

- "Horizontal is the direction in which a perfectly round sphere will not roll on a pefectly flat surface"

 

is not valid since perfectly round spheres do not exist in nature. (apart from possibly bubbles, but they don't roll).

 

- "Horizontal - the direction from a point on the earth directly to the horizon"

 

This is pretty close but the direction and the distance of the horizon depends on how tall you are. If you are 2000m tall (or on top of a mountain/building) your view of the horizon is now in a downward, diagonal direction, not at right angles to the line of gravity (drawing not to scale)

 

 

If you are exactly 0 meters tall, and assume the earth is a perfect sphere, then the horizon will appear 0 meters away, giving you a point, not a line.

 

Ignoring this effect, your view of the horizon is altered by natural features - mountains, pull of the moon on the sea etc. Also, the exact point of the horizon, where the sky meets the sea, is hazy, due to the amount of atmosphere between the horizon and the obsever.

 

Look at this diagram, (not to scale) showing the earth, the sun, and the line of gravitational pull between the two, and show me where the natural right angles are:

 

 

 

Regardless of all this, gravity cannot explain the continuous appearance of the right angle is so many other human endeavours, for instance the right angle grids on a chess board and other game boards.

 

Spatially, the human mind loves to interact with squares, rectangles and right angles.

This is why square chess is massively, massively popular.

Chess played on a hexagonal gameboard, while perfectly valid, remains a marginal activity.

 

Sudoku swept across the world in 2005 and now appears in the backpages of just about every newspaper every day. Hexagonal and triangular versions remain also-rans.

 

Even the children's game of hopscotch belies a fundamental human fixation with right angle.

The purpose of the boxes is to contain the children's feet; feet are not square.

 

A window, the purpose of which is to allow light into a room, could be any shape: circular, hexagonal, nonagonal. But predominantly so, it is a rectangle.

 

A bed, the purpose of which is to contain a prostrate human, is predominantly a (rounded) rectangle. The human body is not rectangular.

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Sayonara:

Don't you think that the efficiency benefits of regular geometry might be a simpler explanation?

 

Mathematics has proved that when packing circular objects to minimize volume used, the most efficient regular geometric shape is the hexagon, not the square.

From my calculations, about 12% more efficient.

 

 

CaptainPanic:

if take a point on that line, and you want to make a second line through that point and have an equal angle to both sides, then the perpendicular line is the one you will find.

 

this one is really just a question of sentence semantics, but here is a straight line, with a second straight line through a point on the line, with equal angles alpha1 and alpha2 on both sides:

 

 

But of course that's not what you meant, and it's a simple matter to adjust the sentence to "cross two straight lines so they create four equal angles", essentially dividing the circle in four like so:

 

 

There is no question that this is simple and correct mathematics.

Regardless of its simplicity however, dividing a circle in four to obtain a right angle remains an algorithm designed by humans, and thus not natural.

 

We could also use a similar algorithm to divide the circle in 6, 7 or 9 equal angles, creating the perfect geometric shapes the hexagon, septagon and nonagon, and label all of those natural:

 

 

The hexagon appears in nature (the beehive); the septagon and nonagon do not (to the best of my knowledge)

 

The bees, were we able to communicate with them, would see our system as needlessly complex. They need not go to a fourth division of the circle. For their perfect system, they need only divide the circle in three:

 

 

creating their perfect right angle: 120 degrees.

 

 

Overall, I am not questioning for a moment the validity, correctness, usefulness or beauty of mathematics or physics. I am questioning the overabundance of the right angle in these systems and in the full array of human physical and theoretical creation.

In mathematics, that questioning begins with the origin itself.

[CaptainPanic]

Bees, if they could make plots would also use 90 degrees. Anyway, if you are really-really convinced that bees would be able to do maths the bee-way, then please move on to translate the theory of trigonometry into this 120 degree world.

 

Humans build staircases which do not use perfect 90 degree angles for the vertical sections. This is to increase the surface where you can put your foot, to be able to have a steeper staircase without losing your footing. I really don't understand what a staircase has to do with the scientific and mathematic meaning of the perfect right angle.

 

Until now, all you have been saying is that it's possible to make projections .

[Sayonara]

Ok thanks for the input (ps. keep it coming!). Before I address your points, i'll throw a couple of curveballs.

Or you could address the points.

[pyxxo]

CaptainPanic:

please move on to translate the theory of trigonometry into this 120 degree world

 

A simple matter. Nothing more than a simple trifle.

 

 

This took no more than two minutes to calculate.

 

You will note that

 

 

where C is the critical angle which is always 90 degrees in standard trigonometry, but now 120 degrees (in radians).

 

Go ahead and do the calculations.

It's a simple matter of measurement of lines, angles and some arithmetic.

 

And if it works for 120 degrees, it will surely work for any critical angle C.

 

Further,

 

 

where the functions on the right of the equation are the normal, 90 degrees versions. If you can simplify that please do.

 

Since

 

TAN A = SIN A / COS A

 

BEE TAN A is trivial

 

Now simply replace SIN/COS/TAN in all the normal trigonometry equations with their bee versions.

 

This doesn't provide for equations which use lengths (laws of sines / cosines etc) but they can probably be converted, if SIN/COS/TAN were converted that easily.

 

Now look at this table of common trigonometric functions

 

 

Notice the preponderance of pi/2?

 

pi/2 is equal to 90 degrees

 

The entire system of trigonometry has to compensate for 90 degrees because it is dependent on 90 degrees.

 

If you use 120 degree triangles, or any other triangles for that matter, simply replace pi/2 with your angle, thus removing all trace of 90 degrees from the system, (assuming all the normal sin/cos/tan tables have been converted.)

 

Now think about this. Hundreds of thousands of mathematicians, physicists, engineers and architects across the world use trigonometry every day of the week in endless calculations. All without a care as to why the system is built on 90 degrees.

 

Nowhere in trigonometry theory does it explain why 90 degrees triangles are used, over any other angle, and no one has sought to ask the question, to the best of my knowledge.

 

With a few simple calculations it can be shown that trigonometry can be performed with any angle.

 

For pure mathematics, there is no rational or logical reason to prefer 90 degree triangles over any other. (for building a right angle dwelling, there is)

 

So, here is the answer:

 

Because the Human Mind has as its fundamental spatial root, the right angle.

 

 

I now declare the use of the right angle in trigonometry superfluous.

 

 

Up to this point, there has been an unshakeable belief in the perfection, correctness and absolute necessity of the right angle in trigonometry. Whereas it is simply one of any useful angles from 0 to 180 degrees.

 

And now, if it was shown in a matter of minutes that the right angle is superfluous in trigonometry, how many other areas of mathematics is it superfluous?

 

 

Once again, I am not for a moment questioning the correctness of mathematics. I am questioning the overabundance of the right angle.

 

 

All of this, still, is only a small portion of how the right angle human mind affects the human-made world.

 

Yes, it feels right. When you play tetris, the way the blocks move in steps, click together and rotate, all in a perfect orthogonal grid - each click feels right - because the human mind prefers 90 degrees over say 60 or 120.

 

Versions designed using the other two regular polygons which stack together without space leftover, the hexagon and triangle, remain sideline activities.

[Kyrisch]

The entire system of trigonometry has to compensate for 90 degrees because it is dependent on 90 degrees.

 

And this is where you inadvertently prove the point that has been the main response to all this that you're spewing all along.

 

Trigonometry wasn't invented by people, its laws and identities were discovered, defined by the actual physical constraints of geometry, something that exists independent of the human mind. The reason that 90º is such a prevalent constant in the identities because it corresponds to a rotation, again relating back to the fact that dimensions are defined (and are so because the math is simplest) to be orthogonal to each other.

 

And furthermore, your logic is faulty due to a huge and obvious fallacy of accident. There are plenty of curves in human engineering, from all sorts of recreational balls to lightbulbs to clocks to knobs to wheels. And you may argue that this is because such things necessitate a spherical shape in order to function -- well, so do all those things composed of right angles! If laptop screens were not square, they would not be able to fold efficiently over; if computer keys were not square, they would have bad packing efficiency; if rooms were not rectangular, it would be difficult to put furniture in them in any efficient way -- flat sides go well against flat sides. Any angle other than 90º (such as 120º) would cause oblique sides and awkward furniture arrangement. Curved surfaces would be even worse. Instead of having randomly shaped furniture to fit that various circumstances arising from a non rectangular room, it's much, much easier to just make everything rectangular and have it all fit nicely wherever one chooses.

[pyxxo]

Someone please run the numbers through those equations I gave previously and tell me I'm wrong

[Kyrisch]

Someone please run the numbers through those equations I gave previously and tell me I'm wrong

 

Your numbers are not incorrect, it's just that you're not drawing a valid conclusion. It's not your math, it's your concept and theory that we are rebutting.

[CaptainPanic]

Because the Human Mind has as its fundamental spatial root, the right angle.

 

Of course the human mind automatically likes the right angle. But that's not because of us, humans. It's because it's such a fundamental angle which occurs everywhere in nature.

 

It's not invented by man.

 

The only thing that was invented by man is to assign the number "90" to the right angle. And in fact, some systems assign pi/2 to it. Or 100% (for slopes of mountains (I think - not too many mountains in my country)).

 

Also, to go back to your 1st post: why are our roads often at 90 degrees? Because that minimizes the road surface, while optimizing the surface for buildings. 90 degrees is not an arbitrary choice. It's a logical choice that follows from calculations/optimizations and also sometimes from nature itself.

[pyxxo]

ok let me explain it another way. Here's 24 cans of beer, packed in a standard crate the normal way.

 

 

Billions and billions of cans of beer make their way around the world every day of the week.

 

Now surely, if there is a constant need for optimization and efficiency, and if land prices for storage come at a premium, surely we would by now have found the best way to pack 24 cans of beer into a box.

 

For simplicity let's call the diameter of a can 1 unit. So the area of the above box is 6 x 4 = 24 units.

 

Now, increase the length of the box by ~3%, also increasing the area by 3%, and pack the cans in a 4-3 hexagonal formation.

 

You can now place the mythical 25th can of beer into the box:

 

 

giving you an extra 4% beer for a space saving of ~1%.

 

Ok, 1%, no big deal.

 

Now do it the other way. Increase the width of the box by ~11.6%, and pack in a 6-5 hexagonal formation.

 

 

Now you can place an extra 4 cans of beer into the box, giving you an extra ~17% beer for a space saving of ~5%.

 

If you use the (less common) 8 x 6 = 48 square packing, you can actually place 2 extra cans in without even changing the shape of the box:

 

 

And that's if you use rectangular boxes. If you use hexagonal boxes, you save 10% on a 19-hex formation, 8% on 7-hex and 11% on 37-hex

 

 

Here's the numbers:

 

 

Now multiply that by every crate of beer in the world and it adds up. Fuel costs for transport would be the same, as the weight of one can of beer is the same, but the land costs for storage would make a difference.

With hexagonal boxes you would lose some efficiency at the edges of the (always rectangular) truck / warehouse they're stored in, depending on its size and shape, but this could be reduced by using hexoid (elongated hexagonal) boxes.

 

The efficiency of hexagonal packing has been known for thousands of years.

But we don't use hexagonal packing. We use square packing.

In this case, not only is there no rational or logical reason to use a square formation, there is a very very good reason not to use it.

 

No law, equation or formula in mathematics or physics can explain this.

 

Square packing of cylindrical objects is used because it fits into the mind of every person in the chain from manufacturing to retail.

[Kyrisch]

Rectangular arrays are easier to count, and the problem of fitting hexagonal boxes into non-hexagonal things would counteract any strictly gained efficiency. Look, you can give all the examples you like, but you still haven't addressed any of our counterpoints.

[The Bear's Key]

When we design, create, build or arrange objects, the most basic, fundamental shape we use is the right angle and its progeny: the rectangle, square, cube and cuboid.

Not always. It depends on the person. But you're right, it can take more advanced skill to build with circles than to use straight lines and corners.

 

Also when people think of futuristic cars, houses, etc, they usually picture imagine curves for elegance and technological greatness.

 

Futuristic design.

 

Comfortable architecture.

 

 

But funny enough (when drawing), our preferred shapes tend to depends on the stage of life we're at. Give a toddler crayons, and we'll often see more circles than right angles.

 

(Visual reminders.

http://www.bartelart.com/arted/wallscribblers.html

http://www.artsz.org/child-art

http://blog.bolandbol.com/drawing-as-it-develops/circles-and-coloring-books-a-mistake

Did you notice how some kids will draw more edges and straighter lines than others? But nonetheless, circular scribbles remain the norm)

 

 

Here's the modern city of Johannesburg, beside the city of Kahun, near Giza in Egypt, built 5,000 years ago:

 

Straight and right angles is cheaper to plan/build, good enough for the masses and heavy transportation, perhaps. Why spend unnecessarily on the lower class?..goes the thinking. Anyone with enough money residing in the little suburbs-type neighborhoods often have lots of bending streets -- no grid, more like a handful of noodles curving randomly with barely any outlets.

 

 

Here's various sports pitches: basketball, football, badminton, snooker, volleyball:

 

We find circular examples too.

 

Roman-ball and hockey

 

 

Duck-Duck-Goose and marbles

 

 

Sumo wrestling, golf, and racetrack events

 

 

Cricket and the Roman Coliseum

 

 

The ancient games seemed to use circles a lot more -- and if true, it weakens your case for inherent bias towards right angles.

 

Again, the masses. Kids learn print before cursive, in Western nations at least -- it might be different in Eastern languages (hopefully we're going back to teaching cursive -- before print -- as it's seeming that children master associated skills better with cursive). Point is, the rigid lines of capitalization might not be an inborn preference, but a holdover font style intended for printing machines which couldn't do cursive.

 

Regardless, I disagree that we inherently prefer right angles. Take a look at the ancient discoveries (the wheel, pi, and fire), variables that history seems to emphasize as more important to our achievments than the right angle.....which happens to be only a partial circle, or 1/4 to be exact -- so not coincidentally, perhaps, all the variables share a trait: curviness. (fire included )

 

 

 

Let's compare humankind's fixation with right angles to that of another animal that also spends its days creating a perfect, regular geometric shape: the bee. It's shape of choice: the hexagon:

 

 

Just like the human-made world is dominated by 90 degrees, so, the bee-created world, the honeycomb beehive, is dominated by the angle at the corner of a hexagon: 120 degrees.

No, it's not.

 

 

I see more right angles than instances of 120 degrees.

 

Bees, if they could make graphs, would make them like this, with 120 degrees as the critical angle:

That's not a fact.

 

Spatially, the human mind loves to interact with squares, rectangles and right angles.

This is why square chess is massively, massively popular.

Women, and their curves, are even more popular to interact with. Curvy sports vehicles too.

 

 

Now surely, if there is a constant need for optimization and efficiency, and if land prices for storage come at a premium, surely we would by now have found the best way to pack 24 cans of beer into a box.

 

 

A much simpler way to maximize efficiency?

 

-- Use square cans. However, it's likely been thought of, and maybe didn't go over too well.

 

I'll leave you with some pics to glance at.

 

 

Contrasts

 

Spherical construction (above) often does look more elegant than right angles.

 

 

...but straight lines and corners look just as (or more) elegant when done good.

 

 

 

Pics of circular stairs

 

http://lefflandscape.com/pix/feature-images/masonry/Circular-steps-sm.jpg

http://www.cityofsound.com/photos/swiss_cottage_library/swisscottage_stairs2.jpg

http://www.scrapbookscrapbook.com/DAC-ART/images/spiral-stairs/spiral-stairs-cambridge-england-1.jpg

http://www.arundelfenceanddeck.com/images/showcase/small_stone%20steps%20(1).jpg

 

In my opinion, those look better than many of their straighter counterparts.

[John Cuthber]

So far as I can tell you can't have an angle ( whether it's 90 degrees or something elde) without havig two straight lines so the simplest unit of spatial thought might be the straight line. But hang on, you can't have a straight line without defining it as the shortest path between two points so the point must be even more fundamental than the line.

 

Alos, on the graph plotting front, using axes at 90 degrees makes use of the fact thatt he two coordinates are orthogonal in the matematical sense- you can vary one without varing the other.

[Mation]

I now declare the use of the right angle in trigonometry superfluous.

Bravo! You are very entertaining

 

on the graph plotting front, using axes at 90 degrees makes use of the fact thatt he two coordinates are orthogonal in the matematical sense- you can vary one without varing the other.

Good point.

 

Don't overstate your case, pyxxo. You ask some really good questions, so it would be a shame if people got distracted by (correctly) rebutting details that aren't integral if the concept is sound.

[lakmilis]

ok... will go back and read all the fancy imagery postd after post 1.. but WHY right angle you say? Well , right angles (orthogonalism is inherently how we could sub divide the prima materia.. the perceivable world.. i.e. the world defined by our 5 exoteric senses given we are not disabled), give us the way to ddivide it further into its 3 constituents... HEY .. welcome mathematics (geometry modelling). Why does it happen to be 90 and not 120 or 213? Well.. this is because of the solar system.. sun and moon... the good ol' Babylonians divided the year into 360 days and also did that to the repeating pattern.. which the circle represented... voila.. out came the 90 degrees being 360/4.

 

Nothing revolutionary there kid, in fact.. if we would meet aliens somewhere... giving ANY description in terms of ' degrees' would be localised to our planet and its solar system... any smart geometer (cheers ajb), would never attempt to explain our angles and divisions in angles in this method, but with PI. or radians... PI is universal.. babylonioan angles are local to tellus.

 

If I told a pink dotted alpha centurian.. hey.. look a circle is divided into 60 degrees.. I would not just have a lingual problem, but a semantic problem as hell.. he would never agree... but if I so neatly started developing our common translation with taking a diameter of a circle.. then putting it around the cirumference and then wrote [MATH] /pi [/MATH] , he would go OH YEAH.. im with you on that one.. we call it donkeyballs (as an example, replace donkeyballs with predatorian sign language if you wish).

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circle divided into 360 sorry. /pi = \pi

[pyxxo]

CaptainPanic:

Bees, if they could make plots would also use 90 degrees.

 

Why would they? Look at the physical world the bees create.

With stunning regularity, over and over, they (and some wasp species) create the angle of 120 degrees. Nowhere do they make 90 degrees.

 

Incidentally, the 90 degrees is also found in nature: It is the angle between a flat surface of water and the direction of gravity. (Agreed, there is some rounding off here - the earth is not exactly round).

 

As mentioned, true right angles so favored by humans, rarely appear in nature. For gravity to be involved there must be a rendering of the line, for instance, a perfectly fluid trickle of water from a stream, or a tree growing upwards. In most of these cases the right angle is negated by such curves as the roots of the tree, the uneveness of the surrounding ground, the surface tension of the water.

The right angle does exist in nature in the form of crystals.

 

The square grid exists in exactly zero places in the entire natural known universe.

This makes it as unnatural as the airplane and the microchip.

 

 

Also, right angles occur in many places in physics (for example the Lorentz force - it wouldn't become 85 degrees if we changed our definition).

Lorentz force: when a wire is moved at right angles through an electric current, the wire will experience a small force at right angles to both.

This depends on the magnetic field moving at right angles to the wire (or vice versa). If you move the magnetic field at a diagonal angle to the wire, the Lorentz force will also be diagonal.

Look at the diagram; simply move the magnetic field (blue B) at an angle to the wire, and the resultant force (red F) will be diagonal.

The orthogonality of the natural force depends on an orthogonal human input.

 

Kyrisch:

And this is where you inadvertently prove the point that has been the main response to all this that you're spewing all along.

I'll explain it another way:

90º trigonometry compensates for 90º because it's dependent on 90º

120º trigonometry would compensate for 120º because it's dependent on 120º

57.296º trigonometry would compensate for 57.296º because it's dependent on 57.296º

120º trigonometry would not compensate for 90º because it's not dependent on 90º

 

 

Calculating sin/cos/tan values is not simple. There is no easy straight formula for calculating sin for an arbitrary angle. The ancient greeks used square root algorithms; computers use infinite series derived from Taylor's theorem. Both are computationally expensive. If you want a lot of precision, you must perform a lot of calculations. There is nothing pretty, beautiful or natural about high-precision infinite series calculation for an arbitrary angle.

 

 

Further to the equations given in my previous posts, rather than 120º a simpler angle to use would be 30º so now sin30(A) = 2 * sin90(A) which removes the square root of the 120º formula.

To calculate sin30(A) to your desired precision, simply multiply the infinite series above by 2.

 

This is far far simpler than the conventional trigonometric method of dealing with a scalene triangle, which is to separate the triangle into two right angle triangles, and perform two expensive calculations.

 

Given an angle and a distance, the cartesian system must use such expensive trigonometric computations in solutions. The polar coordinate system has angle and distance built in as its default variables.

 

 

Kyrisch:

Trigonometry wasn't invented by people

Regardless of how many natural distances, motions or forces trigonometry describes, there is only one place trigonometry, defined as the study of triangles, exists, and that is in the minds of humans.

 

Let me break it down for you:

 

Fire -> exists in nature -> discovery

wheel -> does not exist in nature -> invention

right angle -> exists in nature -> discovery

square grid -> does not exist in nature -> invention

force of gravity - > exists in nature -> discovery

line drawn on paper to represent gravity -> does not exist in nature -> invention

performing trigonometrical calculations on paper -> does not exist in nature -> invention

 

 

Trigonometry is indeed very successful at describing the motions of the planets in cartesian space. Now look at two naturally orbiting bodies, and show me where the right angles are. Circular motions in a 2D plane are better described using polar coordinates.

 

Kyrisch:

The reason that 90º is such a prevalent constant in the identities because it corresponds to a rotation, again relating back to the fact that dimensions are defined (and are so because the math is simplest) to be orthogonal to each other.

 

If you are describing right angle shapes, then yes orthogonal axes are optimum.

If you are describing the circle, which is found in many places in nature (the pupil of the eye), polar coordinates are simpler.

In the right angle world, the equation for the unit circle is an ugly and complex x² + y² = 1

In the polar world, it is simply r(θ) = 1

 

Archimedes spiral, described in the right angle cartesian world, is a cumbersome and lengthy

 

y = x tan ( √ (x² + y²) / a )

 

In polar coordinates it is simply r(θ) = a + bθ

 

And guess what, kids, the spiral appears in many natural formations (spiral galaxies, hurricanes) and in organic lifeforms (snail shells).

Other organic forms such as the helix and basic flower shapes are easier to represent in polar coordinates

 

 

All the massive variation in natural life on earth has produced no true right angles, and no square grid.

[The Bear's Key]

With stunning regularity, over and over, they (and some wasp species) create the angle of 120 degrees. Nowhere do they make 90 degrees.

 

Right here they do. You see it within the honeycombs? The 90s angles outnumber the 120s. The 90s even look slightly more precise.

 

 

Regardless of how many natural distances, motions or forces trigonometry describes, there is only one place trigonometry, defined as the study of triangles, exists, and that is in the minds of humans.

 

Let me break it down for you:

 

Fire -> exists in nature -> discovery

wheel -> does not exist in nature -> invention

right angle -> exists in nature -> discovery

square grid -> does not exist in nature -> invention

force of gravity - > exists in nature -> discovery

line drawn on paper to represent gravity -> does not exist in nature -> invention

performing trigonometrical calculations on paper -> does not exist in nature -> invention

That seems to have merit. However, remember there's a difference between calculations made on paper and the trigonometrical components existing in nature. The only reason it's done on paper is for ease.

 

You're correct it's in the human mind, though. But it couldn't be if the foundation for it didn't exist in nature.

 

Our method of systemizing the calculation of its variables does reside in the mind, but those actual variables do exist in nature, otherwise we couldn't identify them.

[Sayonara]

Pyxxo, I will say it again: Address the points.

 

If all you have is repeatedly stating your opinion and posting pictures then this thread is going to end up being locked. I am moving it into the P&S forum until you provide something more robust than speculation about what sort of graph bees would draw.

[pyxxo]

Sayonara:

Pyxxo, I will say it again: Address the points.

If you look back, you will see that prior to this post, I have quoted and addressed the points no less than 13 times. I will continue to address the other points.

 

If all you have is repeatedly stating your opinion and posting pictures then this thread is going to end up being locked. I am moving it into the P&S forum until you provide something more robust than speculation about what sort of graph bees would draw.

 

Here is some evidence from anthropology:

 

 

- a series of rectangular grids, painted on the wall of a cave in Lascaux in France, 17,000 years ago.

Up to this point, the entire world community of anthropologists have failed to provide any explanation for these symbols.

Feel free to offer your own explanation for this.

The animal figures painted on the wall are explained by existence in nature. Likewise, series of dots have been explained as star maps.

 

These right angle symbols predate mathematics, written language, metal tools, pottery vessels, the wheel and agriculture.

 

As mentioned, the square grid exists in exactly zero places in the entire natural known universe. This shows that the right angle grid was a fixation of the human mind since at least 17,000 years ago, and cannot be explained by any natural component.

 

-------------------------------------------------------------

 

Now, some more of the points:

 

Kyrisch:

And furthermore, your logic is faulty due to a huge and obvious fallacy of accident. There are plenty of curves in human engineering, from all sorts of recreational balls to lightbulbs to clocks to knobs to wheels.

 

Agreed, there are many many curved objects in human creation.

Many of which can be explained by necessity or some natural component.

 

The clock: this comes from the sundial, where the sun's shadow traced a curved line over a day's sunlight. This explains why clocks are 12 hours, even though a day is 24 hours. Further, a clockwork hand traces a circular path.

The wheel / the ball: cubes don't roll. Square wheels don't roll. Sounds obvious? It took the human race 6,000 years of wheel-less toil during the agricultural revolution before we finally figured it out in 3700 BC.

Lightbulb: spheroid due to the vacuum; square or hexagonal glass would be stronger at the corners and weaker at the sides. Further, the corners would produce ugly shadows.

 

The wheel, the clock, and the lightbulb are all late inventions.

The human use of the right angle, and the invention of the square grid predates these by many tens of millennia.

 

And you may argue that this is because such things necessitate a spherical shape in order to function -- well, so do all those things composed of right angles!

This is exactly my point - too many right angle formations keep showing up where they are unnecessary, and/or other shapes are more efficient or suitable: cf. the periodic table, square packing of food/drink cylinders, trigonometry, sports pitches.

 

If laptop screens were not square, they would not be able to fold efficiently over

Triangular, pentagonal, hexagonal, trapezoidal, rhombic, capital-D shaped, or pretty much any shape laptop with one straight edge for a hinge, would fold over just fine.

 

if computer keys were not square, they would have bad packing efficiency

As mentioned, hexagons have the best packing efficiency

 

if rooms were not rectangular, it would be difficult to put furniture in them in any efficient way -- flat sides go well against flat sides.

Hexagons and triangles have flat sides.

 

Any angle other than 90º (such as 120º) would cause oblique sides and awkward furniture arrangement. Curved surfaces would be even worse.

 

Furniture with 120º corners would be fine in a room with 120º. Any angle fits neatly into another object of the same angle. Look at these 'less than' characters: <<

The first thing is that no living space is simply a rectangle packed full to the brim with other rectangles. There are spaces to walk around, and rectangular furniture is regularly arranged so that gaps appear.

Eg. a CRT TV is quite often placed diagonally in the corner (although newer LCDs fit nicely flat on the wall), and 3-seater and 2-seater couches are usually placed at right angles arm to arm, creating a gap in the corner which is harder to access.

The general feel of a living room full of say, 6 people talking, is circular, around the center of the room. People don't naturally line up at right angles to each other when congregating.

 

Non-congruent rectangles don't necessarily tile without gaps, i.e., given a random assortment of rectangles of different sizes and shapes, they may not necessarily pack into a larger enclosing rectangle.

 

-------------------------------------------------------------

 

CaptainPanic:

why are our roads often at 90 degrees? Because that minimizes the road surface, while optimizing the surface for buildings. 90 degrees is not an arbitrary choice. It's a logical choice that follows from calculations/optimizations and also sometimes from nature itself.

 

Çatalhöyük, an ancient settlement in Anatolia dated to 7,000BC, used cuboid bricks and roughly rectangular buildings. Yet it had no streets at all. The buildings were stacked beside each other; the door was a hole in the roof.

 

 

Even that long ago, the right angle building was within the very fabric of human society, before the invention of the street itself.

 

This shows a human fixation with right angle buildings that is completely devoid of the concept of roads.

 

 

Kyrisch:

Rectangular arrays are easier to count

Rhombic arrays count and multiply exactly the same as rectangular arrays.

 

and the problem of fitting hexagonal boxes into non-hexagonal things would counteract any strictly gained efficiency.

Ok, assuming you ignore the hexagonal boxes - and just use hex-in-rectangle boxes. These boxes pack exactly the same into rectangular trucks, warehouses etc.

 

Further to the figures given above, here's the top 10 most efficient hex-in-rectangle packing formations for 48 or less cans:

 

 

So the 28-can 6-5 formation given previously is beaten by a 23-can 8-7 and a 26-can 9-8 with 5.23% and 5.74% respectively better than square packing.

 

If you want smaller boxes, here's the best formations for boxes of 20 or less:

 

 

Again, there is no rational reason for humans not to use the more efficient hex-in-rectangle packing. And, indeed, we already use it to pack 20 cigarattes into a rectangular box in a 7-6-7 formation, which is 4.58% more efficient than square packing.

 

The industry doesn't seem to have any trouble counting those hexagonal arrays.

 

Go ahead and explain why the food/drink industry does not take advantage of this efficiency.

Edited April 15, 2009 by pyxxo
added line at the end

[MM6]

Consider the common staircase - which performs a simple function - the vertical transport of humans:

 

 

The horizontal portion, the tread, is horizontal for good reason - to avoid slipping. But the vertical portion, the riser, has no reason to be perfectly vertical.

The vertical riser could be slanted in either direction, like this:

 

 

The first case, the outward slant, is less safe than the standard right angle due to a smaller horizontal surface.

By that same logic, the second case, the inward slant, would be safer than the right angle version.

Yet across the world, standard staircase design remains mostly perpendicular.

 

Not because it's the best design. But because its the easiest design for humans to create and build.

 

A perpendicular support experiences less shear force than the slanted supports. It also uses less material. It's a better engineering design.

 

That staircase image caught my eye so I'm picking on that one example. I haven't read your entire post(s). You've obviously put a lot of thought into this. I'm looking forward to reading it.

[John Cuthber]

I have seen plenty of stairs which are more or less like the second case shown.

It's common for the riser to be set back from the edge of the step when using timber.

Like these.

http://www.blocklayer.com/stairs/

But with concrete the design is pretty much pyxxo's second diagram.

http://images.google.co.uk/imgres?imgurl=http://www.builderbill-diy-help.com/image-files/st-concr.jpg&imgrefurl=http://www.builderbill-diy-help.com/formwork-stairs.html&usg=__VMrFpT-1CjqSBKn_UGXlOccuk7o=&h=364&w=484&sz=24&hl=en&start=16&um=1&tbnid=usDrpgKF9xWl9M:&tbnh=97&tbnw=129&prev=/images%3Fq%3Dconcrete%2Bstairs%26hl%3Den%26rlz%3D1T4GGLJ_en%26um%3D1

 

Also, a bee hive has a stack of hexagonal tubes, but they have a nearly flat face. That face is at right angles to the axes of all the tubes. There are, therefore, plenty of right angles in bee hives.

[MM6]

I have seen plenty of stairs which are more or less like the second case shown.

 

That's true b/c the concrete stairwell is a solid block, in which case shear force is nullified. It's all compressive force, which is where the true strength of concrete lies.

 

I was considering the cases in which the interior is hollow and the structure is built with wood. Like the stairs in/around most homes.

 

Good point, though.

---

Merged post follows:

Consecutive posts merged

In the video I make the statement:

 

Apart from right angle crystals, there are no true, natural right angles in the known universe.

 

pyxxo,

 

What exactly is your hypothesis? That's what I'm looking for, a simple statement so that I know what we're arguing for or against.

 

Anyway, I can think of plenty of right angles that exist in nature. Here are two categories.

 

1) Right angles occur in chemical bonds

2) There are several perpendicular relationships among charged particles passing through magnetic fields.

 

Grids in nature. I'd have to think more about this one.

 

Reiteration: Could you please state your hypothesis?

Edited April 19, 2009 by MM6
Consecutive posts merged.