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

[pyxxo]

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

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.

 

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.

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°.

 

 

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.

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.

 

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

1) Right angles occur in chemical bonds[/quote']

A right angle is defined as two straight lines touching at 90°. Any lines drawn in a molecular diagram remain simply that; imaginary lines drawn by humans. The electromagnetic force between the atoms is real; the line is not.

No line, no right angle.

 

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

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.

[Sisyphus]

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.

 

No, you couldn't. He's talking about real, physical forces, that have nothing to do with how we define them. Gravity would still obey the inverse square law, even if The Great Definition Authority decreed that it was instead an inverse cube. For that matter, it would still act at right angles to the surface, which happens to be the reason concrete stairs are perpendicular. It is not arbitrary, and is not the result of something peculiar to human minds. It is the result of physics.

 

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 force results from the perpendicular component of the motion.

[pyxxo]

No, you couldn't. He's talking about real, physical forces, that have nothing to do with how we define them.

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.

 

Gravity would still obey the inverse square law ... For that matter, it would still act at right angles to the surface

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.

 

which happens to be the reason concrete stairs are perpendicular.

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.

 

The force results from the perpendicular component of the motion.

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.

[Kyrisch]

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.

 

If the "input" is "diagonal", the resulting force is still perpendicular with respect to the first vector, unconditionally.

[MM6]

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)

 

My goodness. Then take astronomical scales where the surface topology becomes irrelevant. The earth's pull on the sun or on any body on the other side of the galaxy occurs on a line to it's center and at a right angle to the surface of the earth.

 

As Sysiphus and Kyrisch have followed up on my points, which you failed to refute in any sensible way, I'm done here.

 

You appear so wedded to your hypothesis that you're unwilling to accept any evidence that may refute it. That's ideology, not science.

[pyxxo]

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.

 

 

Then take astronomical scales where the surface topology becomes irrelevant. The earth's pull on the sun or on any body on the other side of the galaxy occurs on a line to it's center and at a right angle to the surface of the earth.

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.

 

If the "input" is "diagonal", the resulting force is still perpendicular with respect to the first vector, unconditionally.

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?

 

 

You appear so wedded to your hypothesis that you're unwilling to accept any evidence that may refute it. That's ideology, not science.

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

[Sayonara]

I don't suppose it ever occurred to you Pyxxo that packing efficiency is not the sole determinant in deciding what shape containers beer will be stored in?

[Bignose]

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.

 

It is all covered by the Law of Cosines: [math]c^2 = a^2 + b^2 -2ab\cos \theta[/math]. It doesn't matter what angle is chosen, all governed by the same equation. Pythagoras' is just when [math]\cos \frac{\pi}{2} =0[/math]. That's it.

[pyxxo]

I don't suppose it ever occurred to you Pyxxo that packing efficiency is not the sole determinant in deciding what shape containers beer will be stored in?

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?

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It is all covered by the Law of Cosines: [math]c^2 = a^2 + b^2 -2ab\cos \theta[/math]. It doesn't matter what angle is chosen, all governed by the same equation. Pythagoras' is just when [math]\cos \frac{\pi}{2} =0[/math]. That's it.

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:

 

nature is not using the strictly imaginary, awkward, and unrealistic coordinate system adopted by and taught by present-day academic science.

Nature is not employing the three dimensional, [perpendicular], parallel frame of the XYZ axial coordinates of academic science

 

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°.

[Kyrisch]

You still have not commented on the fact that magnetic forces are always perpendicular to the velocity of the charged particle and the direction of the field.

[pyxxo]

You still have not commented on the fact that magnetic forces are always perpendicular to the velocity of the charged particle and the direction of the field.

I did, Read my post #31.

[Sisyphus]

You still have not commented on the fact that magnetic forces are always perpendicular to the velocity of the charged particle and the direction of the field.

 

Or any force with perpendicular relationships, for that mater. Which is to say, pretty much any. Perpendicular motion is always unaffected by the application of any force. Hence horizontal vs. vertical. pyxxo seems to deny this, insisting such things are merely artifacts of definition.

 

Or the more fundamental fact of simple multiplication. Length, width, and depth are perpendicular dimensions, and any none-perpendicular coordinate system, though possible, will greatly complicate the math of physical description. Like with those carbon nanotubes - ok, so how long is the tube, and how thick?

[Sayonara]

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?

I am talking about kegs, of course, the packaging to which you referred in the quote I posted.

 

Anybody who has worked with beer kegs, either at the production end or the service end, will tell you they need to be rolled.

 

I am however all for hexagonal cans, although they would need to be a bit more slender to allow a good fit with the human hand.

[pyxxo]

I am talking about kegs, of course, the packaging to which you referred in the quote I posted.

Anybody who has worked with beer kegs, either at the production end or the service end, will tell you they need to be rolled.

 

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.

[Sayonara]

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.

I think the argument against that is that the barrels are stored on rails so they can be rolled in and out of the transport.

 

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:

That's how cans are packaged in every supermarket I have ever been in

[pyxxo]

I think the argument against that is that the barrels are stored on rails so they can be rolled in and out of the transport.

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.

 

That's how cans are packaged in every supermarket I have ever been in

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.

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Or the more fundamental fact of simple multiplication. Length, width, and depth are perpendicular dimensions, and any none-perpendicular coordinate system, though possible, will greatly complicate the math of physical description

 

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.

 

Like with those carbon nanotubes - ok, so how long is the tube, and how thick?

 

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.

 

Or any force with perpendicular relationships, for that mater. Which is to say, pretty much any. Perpendicular motion is always unaffected by the application of any force. Hence horizontal vs. vertical. pyxxo seems to deny this, insisting such things are merely artifacts of definition.

 

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.

Edited May 14, 2009 by pyxxo
Consecutive posts merged.