## Saturday, December 20, 2008

### What if... #20

What if we could arbitrarily rescale the size of all objects?

This post is part of the 2008 advent series "What if..."

Phil Warnell said...

As Galileo demonstrated
in his work Two New Sciences
that to simply rescale objects or things will not leave them with their original functionality when it comes to strength or durability. If one tried for instance to increase a person’s size to 30 ft tall simply by rescaling his legs wouldn’t be able to support his own weight. Similarly that insect that appears to have such great strength and ability in relation to its size wouldn’t be so considered or able if scaled up to that of a persons height while maintaining its own proportions. The answer is confined to and explained in the relationship of how area relates to volume.

Phil Warnell said...

Just for those that don’t like to follow links below I offer a quote from Galileo’s “Two New Sciences” that most simply explain this as Salviatio explains to Simplicio :

“Take, for example, a cube two inches on a side so that each face has an area of four square inches and the total area, i.e., the sum of the six faces, amounts to twenty-four square inches; now imagine this cube to be sawed through three times [with cuts in three perpendicular planes] so as to divide it into eight smaller cubes, each one inch on the side, each face one inch square, and the total surface of each cube six square inches instead of twentyfour in the case of the larger cube. It is evident therefore, that the surface of the little cube is only one-fourth that of the larger, namely, the ratio of six to twenty-four; but the volume of the solid cube itself is only one-eighth; the volume, and hence also the weight, diminishes therefore much more rapidly than the surface. . .You see, therefore, Simplicio, that I was not mistaken when . . . I said that the surface of a small solid is comparatively greater than that of a large one.”

Although the insight is original the devise of the dialogue used to explain was borrowed from Plato.

Best,

Phil

Phil Warnell said...

It is also interesting to note that Galileo considered the study of materials and scales more important then his astrological interests. With today's realization of the importance and difference of things on the quantum level one must marvel at his insight. You compound this with his first raising the whole subject of relativity to a new level and scrutiny and it’s not hard to imagine why Galileo and not Newton is regarded by many to be the father of modern science.

Although Galileo’s” Discourses and Mathematical Demonstrations Relating to Two New Sciences “ may seem to many today as being antiquated and not in touch with modern thought, I have found it to be the opposite, as it lends one a better sense of the foundations and fundamentals involved.

Have you for instance ever wondered what he might have thought about your own theories and contentions as they relate to minimum length?

cody said...

Richard Feynman discussed the way scaling would influence functionality in his talk, "There's Plenty of Room at the Bottom", and at some point I read a website that discussed the scientific shortcomings of movies which pointed out that if you shrunk normal objects to a much smaller size by removing some of the empty space in the atoms, you would end up with enormous densities. I think they estimated that shrinking a human to a few inches tall would produce such enormous pressures through their feet that concrete would not hold, so I guess we would have to scale things while preserving density?

Rae Ann said...

I think people have always had the desire to try to rescale things. Look at all the "miniature" collectible things they sell like houses, furniture, dolls, cars, and so on. It is part of our human nature to manipulate things - construct and deconstruct. But it isn't only with tiny miniatures but also with giant things too. Maybe this is part of the origin or physics itself?

Arun said...

I'd be quite happy if I could arbitrarily rescale some objects. The most common two that come to mind are to be able to rescale the car for parking, and to rescale telephoto lenses for easy transport.

---

What if Stefan and Bee prematurely end their "What if..." series?

Phil Warnell said...

Although rescaling involves more then simply changing a things relative size, being able to do it without consequence could be very advantageous in regards to our dwindling resources as it relates to our ever increasing population. This could also serve as the title and theme of a movie sequel called “Honey, I shrunk the populous” . :-)

Phil Warnell said...

Hi Arun,

“What if Stefan and Bee prematurely end their "What if..." series?”

I wouldn’t think that should represent to be “adventageous” :-)

Uncle Al said...

Nature is wildly non-linear. One rescaler would be disaster. As for more than one... paddle faster, I hear banjos.

Catadioptric telephoto lenses are small for a given focal length at an f-number. All the fun is in the loopholes! Do left and right shoes vacuum freefall non-identically in WeitzenbĂ¶ck spacetime? Somebody should look.

Arjen Dijksman said...

That would be a big mess. Everyone would try to make his house and garden bigger to the detriment of their neighbours'.

stefan said...

This has happened already in the remote locations of Lilliput and Brobdingnag, according the 18th century reports by an Englishman called Gulliver.

But it seems that today the credibility of these reports is vigorously challenged ;-), because the metabolism of living organisms is related to the ratio of volume versus surface, which varies under scale change.

For example, mice cannot be much smaller than they actually are, and elephants not much bigger, unless they would have to be very different animals. Trees seem to have a maximal height. Hence, the world of complex organisms would look quite different under a general scale change.

On a more fundamental scale ;-), the typical size of objects in the material world is governed by the fundamental constants of physics, as pointed out for example by Victor Weisskopf in "Of Atoms, Mountains, and Stars: A Study in Qualitative Physics" (PDF). And even the mass of the proton, taken as a constant by Weisskopf, is set by the conformal anomaly of QCD, which sets a length, or mass scale... Does anyone know an introductory paper that discusses this issue, a kind of an update to the Weisskopf paper?

Cheers, Stefan

stefan said...

Hi Phil,

Hi cody,

thanks! Here is a transcript of Feynman's talk. I had not read it before - it is very interesting and prescient indeed!

Cheers, Stefan

Phil Warnell said...

Hi Stefan,

Thanks for the pointer to Weisskopf’s paper as it will prove interesting what has been added to Galileo’s original insight. As for the limits of the smallness of mammals although not actually mice Shrew are often as small and physically similar and yet as Shakespeare reminded are indeed fearsome at times and impossible to tame. If these were scaled up to even the size of house cats we would all run for cover :-)

Best,

Phil

Phil Warnell said...

Hi Stefan,

After a cursory look at the Weisskopf’s paper you pointed to which brings much of this together within the realm of current understanding it is realized he doesn’t attempt to apply the rules to the size of galaxies and more importantly to the universe as a whole. It then leaves one to wonder if indeed physical laws place ultimate bounds on the size of a universe.

This was contemplated and eluded to in the past by Newton where in realizing gravity would bring everything together could only imagine that things where so distantly and sparsely spaced that it held no concern. Then we had Einstein who balanced it off with an ad hoc extra term to be later considered again valid yet more concerned with its dissolution rather then collapse. So scale, its limits and the consequences still form to be an ongoing mystery and question. I believe thus that Galileo would be if anything amused.

Best,

Phil

Plato said...

Opticks was Newton's second major book on physical science. Even if he had not made his better-known discoveries concerning gravity and the invention of the calculus, Opticks would have given him the reputation as one of the greatest scientists of his time.Opticks

I know one of science mind would tend more to Galileo, but I would contend that Newton recognized by "a basis" an "alchemical relation" in defining scale. Some might have tended toward the Octave itself and all things an expression thereof. So scale then, was possible. "String tension" and "weight correlation struck" in gourds of water filled and half filled?

The Errors & Animadversions of Honest Isaac Newton by Sheldon Lee Glashow

Now indeed "character lists all the things that science does not require" that one may have an opinion of any man, but truly, what was fostering "the science mind" if he did not see patterns that existed in nature and did not require this assertion of his character?

Newton's Translation of the Emerald Tablet

It is true without lying, certain and most true. That which is Below is like that which is Above and that which is Above is like that which is Below to do the miracles of the Only Thing. And as all things have been and arose from One by the mediation of One, so all things have their birth from this One Thing by adaptation. The Sun is its father; the Moon its mother; the Wind hath carried it in its belly; the Earth is its nurse. The father of all perfection in the whole world is here. Its force or power is entire if it be converted into Earth. Separate the Earth from the Fire, the subtle from the gross, sweetly with great industry. It ascends from the Earth to the Heavens and again it descends to the Earth and receives the force of things superior and inferior. By this means you shall have the glory of the whole world and thereby all obscurity shall fly from you. Its force is above all force, for it vanquishes every subtle thing and penetrates every solid thing. So was the world created. From this are and do come admirable adaptations, whereof the process is here in this. Hence am I called Hermes Trismegistus, having the three parts of the philosophy of the whole world. That which I have said of the operation of the Sun is accomplished and ended.

Use then those most circuitous routes of the brain to foster knowledge with that greater "tentacle of mind" for remembering and far reaching. What we are all made up of, and it was always there for us to ask or wonder about? Prepare the next step.

Best,

Plato said...

Phil,

Between which pages?

Best,

Phil Warnell said...

Hi Plato

"Between which pages?

Sorry a correction it should read:

I guess you are asking between what pages I quoted from "The Two New Sciences". If you use the last reference to the book I pointed to it would be page 83. If you use the one you have pointed to its near the end of the first day on page 90.

Best,

Phil

Plato said...

Thanks Phil.

"How can a speck of a universe be physically identical to the great expanse we view in the heavens above?"The Elegant Universe, Brian Greene, pages 248-249

It goes on to talk about distances further on that page. One can be completely unaware of the other possibility.

I just think it strange that no such thought had been given to the context of energy in this image as a backdrop to this image

Best,

Phil Warnell said...

Hi Plato,

To be honest my copy of “The Elegant Universe” is no longer elegant as it is in tatters:-) What Greene is suggesting is that distance and scale relates more to how we measure them or rather with what we use to compare then having any true significance, so that the speck as it relates to the vastness of a Universe hold no true meaning. To be honest I was never convinced by his argument for what it boils down to is to say that because both infinitely large and infinitely small are infinite they serve to be one in the same or rather that scale is an illusion. I would say that infinity is the limit of either direction and in so being opposite to begin with this can’t be ignored as to be trivialized.

Best,

Phil

Plato said...

While string theory does not, at this point, predict our world, it can at the very least plausibly encompass it. No other theory has been shown to do that. Aaron Bergman's book review of Peter Woit's

Quantifying the extent of it's curvature?

"But now, almost a century after Einstein's tour-de-force, string theory gives us a quantum-mechanical description of gravity that, by necessity, modifies general relativity when distances involved become as short as the Planck length. Since Rienmannian geometry is the mathematical core of general relativity, this means that it too must be modified in order to reflect faithfully the new short distance physics of string theory. Whereas general relativity asserts that the curved properties of the universe are described by Rienmannian geometry, string theory asserts this is true only if we examine the fabric of the universe on large enough scales. On scales as small as planck length a new kind of geometry must emerge, one that aligns with the new physics of string theory. This new geometry is called, quantum geometry." Pg 231, 232

On a everyday level sending perspective out to the "Powers of Ten" we tend to loose capabilities of the larger perspective, yet, it is still possible even though we "hand write the equations" to help perspective to be drawn "about the world we live in."

See: Background and genesis of topos theory

For me it is amazing how such a valley can be descriptive of what the "genus figure" actually represents, and here we could be talking about "one universe?"

There is a "mathematical progression of the genus figures" offered by Mandelstam.

For me those spaces( can we call then this) represent more then the "distant measure" of what we might allot to it." It holds a "spirit of a kind to me" that allows us to easily settle "abstractly" loosing sight of the "greater complexity" as we descent into the matter states? Some contend with "infinity" and lost their way?:)

Best,

Plato said...

Proper image of...how such a valley.

Neil' said...

Like other commenters here I think we could tell if everything was scaled up or down because it is not just a simplistic matter of comparing size alone. For example, make the scale 0.5x but keep charge and laws the same. Then the force is 4x at what is thought same distance as before, which we can measure or see via acceleration. Indeed the acceleration seems even more due to size change so it goes to 8x.

Oddly most philosophers say, we couldn't tell an encompassing size change - maybe they are focusing on the core definition, maybe they mean the entire "scale" changing like enlarging a TV screen so the events would be literally the same. But that just brings up sticky questions of what "scale" ought to mean to begin with - the scaled movie scenario contradicts the "same laws but different size per se" model used above.

Also, if one believes the universe is "made of math" or "it from bit" then there is another concept to changing scale: rescaling the configuration on the "Platonic coordinate system" used to characterize a given conceptual model universe (none more "real" than any other, with no real "materiality" as a distinction.) That PCS would not correspond to any of the arbitrary units we use or even natural units like Planck length, it would just be the numbers that represented the distribution of modeled elements, the locus of their surfaces etc (as it would be in a computer graphics program.) There would be no operational way to tell that the sphere once designated with Platonic diameter 1.5 was now Platonic diameter 3.