# pyxxo

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

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° human-made 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 human-made 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 human-made 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 $\frac{\sqrt{3}}{\pi}$ 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.
2. ## The simplest unit of spatial thought ... is the Right Angle

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

I did, Read my post #31.
4. ## The simplest unit of spatial thought ... is the Right Angle

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°.
5. ## The simplest unit of spatial thought ... is the Right Angle

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 6-5 formation = 94 kegs, for a length of 14.86 keg-diameters. 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 15-14 formation = 102 kegs for an extra 9.75% beer over square packing. How's that for science, MM6? Contrary to what I said above, superior-efficiency 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 all-orthogonal 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
6. ## The simplest unit of spatial thought ... is the Right Angle

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 super-strong, super-lightweight geodesic dome. John Cuthber actually linked to an non-perpendicular, inward-sloping concrete staircase a few posts ago. This shows that non-right 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 many-angled inputs and outputs.
7. ## The simplest unit of spatial thought ... is the Right Angle

Since the concept of "shear" itself is defined as non-perpendicular, 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 ~9-14° 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.

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

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 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: 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.
10. ## The simplest unit of spatial thought ... is the Right Angle

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

Someone please run the numbers through those equations I gave previously and tell me I'm wrong
12. ## The simplest unit of spatial thought ... is the Right Angle

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 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.

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

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. .... .... .... .... .... 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.
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