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The simplest unit of spatial thought ... is the Right Angle
pyxxo replied to pyxxo's topic in Speculations
I've watched trucks unload kegs into bar basements. One worker stands on the truck bed, manually lifts the keg over the side and passes it to a second worker on the ground, who doesn't fully take the weight but lets it drop slowly onto a sand bag on the ground. Even if the truck has the rails you describe, the kegs are upright and would have to be turned on their side to be rolled, which couldn't be done unless the surrounding kegs were removed. So hex formation packing would also work fine with such rails. I've dealt with this in detail above, with diagrams and calculations. I noted that the 48 box was far less common. I reposted it in order to show how, for certain formations, you can place extra bottles / kegs in the box without changing its shape/size. So for a keg truck, 15 x 6, you can fit in 4 more without changing its shape. Like this: From a mathematical point of view, this is absolutely unquestionable. Here's the simple answer as to why the beer industry uses inefficient square formation keg packing: they don't realize that hex packing is more efficient. Merged post follows: Consecutive posts merged This is a perfect illustration of the cartesian mindset. Because we have all spent our lives inside a 90° humanmade world, it's difficult to see past it, or how any other system would work. We view length, width and depth as true, correct and natural dimensions, and it's easy to see how some people can view this as the only valid system. However, as Alfred Korzybski wrote in 1931, The map is not the territory. Spending half a millennium using cartesian coordinates to describe nature does not make nature orthogonal. There is no strict connection between multiplication and right angle dimensions. Placing 4 squares up and 6 squares across is simply one of an infinity of ways to graphically represent multiplication. Multiplication, at it most basic level, is simple repetitive addition, and can be graphically represented using rhombi or parallelograms, once you use the correct unit of area. In this system (which again is only one of an infinity of possibility), length, width and depth are 60° axes. Area and volume calculations work fine. If you are presented with a humanmade square grid or a cuboid, then yes, cartesian multiplication and dimensions are simplest. But given a natural shape such as a leaf, a snail shell or a sunflower head, length, width and depth are pointless dimensions, unless of course you wish to place it into a humanmade 90° cardboard box. For a snail shell and the sunflower head, or indeed the fractal forms of a romanesco broccoli, length and spiral pitch would be more useful. Length  the length of the tube is the same as the number of atoms it has in a line along the length (presuming you can count the atoms!) Thickness  count the number of atoms around, then multiply by [math]\frac{\sqrt{3}}{\pi}[/math] If you would like to describe the structure of a sheet of graphene, the strongest material ever tested, 200 times the breaking strength of steel, again a 60° rhombic coordinate system simplifies the calculations. I haven't mentioned perpendicular motion. I have to stress here, I'm not for a moment questioning the validity or correctness of any existing formulae or relationships, simply that the perpendicular version is only one way to look at it. Let's be specific: what are you referring to when you say 'perpendicular motion'? I will say this: many vector and force equations involve the cross product, which is a resultant vector perpendicular to two given vectors. So the cross product is an abstract operation with no direct component in the natural world, i.e., you can't point at a falling leaf or two orbiting bodies and say yes, look, there is the cross product. Conversely, you can look at a perfectly fluid trickle of water from a stream and say yes, look, that line is a rendering of the direction of the force of gravity. The cross product can be used to correctly describe the forces involved. But it is simply one of many ways to perform the calculations. Just like you can perform multiplication on a 60° rhombic grid if you so choose, you can also define a 60° cross product. 
The simplest unit of spatial thought ... is the Right Angle
pyxxo replied to pyxxo's topic in Speculations
Ah. Good job I asked. I'm talking about leaving the kegs as they are, and simply arranging the many kegs in a hexagonal formation in the back of the truck. In other words, no modification is needed to the kegs or the trucks. Just a small modification to the mental program of the people who load the truck. On a smaller scale, say a box of 48 round objects packed in a square formation of 6 x 8, you can place 2 extra objects into the box without changing its shape, giving you 50 in a 48 box: For the keg truck of 15 x 6, you can fit in an extra 4 kegs. The bigger the box, the more objects you have, the more efficient hexagonal formation packing is. 
The simplest unit of spatial thought ... is the Right Angle
pyxxo replied to pyxxo's topic in Speculations
I did, Read my post #31. 
The simplest unit of spatial thought ... is the Right Angle
pyxxo replied to pyxxo's topic in Speculations
Are you asking about the shape of the secondary packaging (i.e., the cardboard box), the primary packaging (the shape of the beer can/bottle itself), or the formation in which they're packed (i.e., square or hexagonal)? What other determinants do you have in mind? Merged post follows: Consecutive posts merged Correct. So pythagoras shows a special quality of right angles, which I accept as evidence against my case. But as mentioned, 60° also has special qualities, in particular the simplicity of a = b = c for all equilateral triangles. It was Buckminster Fuller who wrote that 60° triangles and tetrahedrons are much closer to how nature is arranged, rather than 90°. He noted: A 60° coordinate system, rather than the standard 90° cartesian system, would make it easier to describe the structure of various molecular structures such as carbon nanotubes (which belong to the class Fullerene, named after Buckminster Fuller) A carbon nanotube is a similar to a hexagonal grid (like chicken wire) wrapped around a cylinder. Since hexagons are arranged around 60° and 120°, if you use a symmetrical rhombic or diamond coordinate axes at 60°, this can define every point in the hex grid. Every point is one unit distance from it's six neighbours. For area, you would use 1 unit of area = 1 equilateral triangle or 1 rhombus. So the area of each hexagon is either 6 or 3, respectively. The horizontal axis is given as y = 0. The 60° axis is given as x = 0. In cartesian coordinates, you would need to use square root formulas to describe the points. In many cases, cartesian is the best system. But it's not the only system. It's my belief that the overuse of the right angle system is a product of the preferences of the human mind, and not the objective efficiency of 90°. 
The simplest unit of spatial thought ... is the Right Angle
pyxxo replied to pyxxo's topic in Speculations
Here's another example of inefficient beer packing, this time on a larger scale: a beer truck filled with kegs in a square formation. That's 15 x 6 = 90 kegs. With hexagonal packing you could fit in 17 rows in 65 formation = 94 kegs, for a length of 14.86 kegdiameters. That's an extra 4 kegs or an extra 235 liters  all without changing the shape of the truck. A modification to the width of the the truck load would allow 7 rows of 1514 formation = 102 kegs for an extra 9.75% beer over square packing. How's that for science, MM6? Contrary to what I said above, superiorefficiency hexagonal packing would save on fuel, depending on the ratio of the truck weight to the load carried. An increased load adds weight thus increasing fuel usage, however this is offset by the weight of the truck itself. I've covered this numerous times above. At the astronomical scale, the earth is curved. The right angle you speak of is a tangent to the surface of the earth, and exists only as an imaginary line drawn by humans. Such a tangent does not exist in nature. If you know of one, please list it here. I consulted an electronic engineer on this and was told the force would be diagonal; could you point to a source on this showing a diagonal wire vs magnetic field vector angle and the resulting force, other than the usual wikipedia diagrams which show only the standard allorthogonal version? I'm happy to accept evidence against my case. Natural right angle crystals being the top of that list. But I also reserve the right to discuss each piece of evidence presented; as indeed you have discussed the evidence I have presented. Another piece of evidence against it is pythagoras' theorem, which is undoubtedly a special property of right angles. As everyone knows, lengths of 3, 4 and 5 make a right angle triangle. But far far less people know that 3, 5 and 7 make a 120° triangle, and 3, 7 and 8 make a 60° triangle: For 90° triangles with sides less than 100, there are 50 integer triangles (aka pythagorean triplets). For 120°, there are 42 integer triangles. For 60°, there are 70 integer triangles. If you add equilateral triangles, add another 99. So, 60° triangles with all integer sides are more numerous than 90° integer triangles, with sides < 100. The formulae for 60° and 120° triangles, c² = a² + b²  ab and c² = a² + b² + ab respectively, are one step more complex than pythagoras, c² = a² + b². However, the simplicity and beauty of pythagoras is beaten by the beauty and simplicity of the formula for the sides of an equilateral triangle, which is a = b = c, and works for the set of all positive integers, and indeed the set of all positive real numbers. The first few 120° integer triangles: 3 5 7 5 16 19 6 10 14 7 8 13 7 33 37 9 15 21 9 56 61 10 32 38 11 24 31 11 85 91 The first few 60° integer triangles: 3 7 8 5 7 8 5 19 21 6 14 16 7 13 15 7 37 40 8 13 15 9 21 24 9 61 65 10 14 16 
The simplest unit of spatial thought ... is the Right Angle
pyxxo replied to pyxxo's topic in Speculations
Yes, you could. Shear stress is defined as being parallel to the surface. Normal stress is defined as being perpendicular. So it would be a simple matter to define a 30° shear. How the relevant formulas would have to be modified is another matter. If and only if that surface is truly horizontal. The majority of natural surfaces are haphazard or rounded with the curvature of the earth. I would go so far as to say that a truly horizontal plane or line does not exist in nature (at least that we know of on earth) Truly vertical lines are rare in nature; one example being a perfectly fluid trickle of water from a spring. The majority of human architectural construction begins with the creation of horizontal surface and usually builds at right angles to it. A notable exception being the superstrong, superlightweight geodesic dome. John Cuthber actually linked to an nonperpendicular, inwardsloping concrete staircase a few posts ago. This shows that nonright angle stairs are perfectly plausible, and in use, and in the case of a solid staircase, use less material. My point is that when we think of a staircase, we usually assume a right angle design. If and only if there is a human induced right angle input into the system. If your input is diagonal, the resultant force is diagonal. The Lorentz force is always described in this right angle format. Why? Because that's the way the human mind loves to view systems. Whereas the right angle Lorentz force is simply one of manyangled inputs and outputs. 
The simplest unit of spatial thought ... is the Right Angle
pyxxo replied to pyxxo's topic in Speculations
Since the concept of "shear" itself is defined as nonperpendicular, the point is moot. You could similarly define a 30° shear and say a 30° riser experiences less 30° shear. Either way, the stringer board (the part on either side of the staircase which holds it together) is at roughly 45°, so for minimum shear the riser would be at 45°. As for material, true, in a hollow wooden design a perpendicular riser minimizes material. But for concrete or any solid material, inward sloping reduces material use. The bees create the cells of the beehive to tilt upwards at ~914° in order that honey / pollen / grubs don't fall out. So the external faces don't sit at 90°. It's in the title of the post. It can be expressed in many ways: The human mind has an unnatural fixation with the right angle. The human mind sees the right angle (and square/rectangle etc) as the most perfect, correct and true shapes, even where another shape would suffice or is more suitable. Like 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. 
The simplest unit of spatial thought ... is the Right Angle
pyxxo replied to pyxxo's topic in Speculations
Sayonara: 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. 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: 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 wheelless 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. 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. Triangular, pentagonal, hexagonal, trapezoidal, rhombic, capitalD shaped, or pretty much any shape laptop with one straight edge for a hinge, would fold over just fine. As mentioned, hexagons have the best packing efficiency Hexagons and triangles have flat sides. 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 3seater and 2seater 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. Noncongruent 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: 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: Rhombic arrays count and multiply exactly the same as rectangular arrays. Ok, assuming you ignore the hexagonal boxes  and just use hexinrectangle 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 hexinrectangle packing formations for 48 or less cans: So the 28can 65 formation given previously is beaten by a 23can 87 and a 26can 98 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 hexinrectangle packing. And, indeed, we already use it to pack 20 cigarattes into a rectangular box in a 767 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. 
The simplest unit of spatial thought ... is the Right Angle
pyxxo replied to pyxxo's topic in Speculations
CaptainPanic: 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. 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. 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: 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 highprecision 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: 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: 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 simplest unit of spatial thought ... is the Right Angle
pyxxo replied to pyxxo's topic in Speculations
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 43 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 65 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 19hex formation, 8% on 7hex and 11% on 37hex 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. 
The simplest unit of spatial thought ... is the Right Angle
pyxxo replied to pyxxo's topic in Speculations
Someone please run the numbers through those equations I gave previously and tell me I'm wrong 
The simplest unit of spatial thought ... is the Right Angle
pyxxo replied to pyxxo's topic in Speculations
CaptainPanic: 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 humanmade 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. 
The simplest unit of spatial thought ... is the Right Angle
pyxxo replied to pyxxo's topic in Speculations
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 sansserif 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 lesserused 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 humanmade world is dominated by 90 degrees, so, the beecreated 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. Merged post follows: Consecutive posts mergedOk, now to your points Sisyphus: 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. Again, lines at 60, 30, 78.34 degrees etc, also have extension in different dimenions. CaptainPanic: 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: 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 alsorans. 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. Merged post follows: Consecutive posts mergedSayonara: 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: 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. 
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, tictactoe, 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. .... .... .... .... .... 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.