"...Why is symmetry so satisfying?"

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Looks like an interesting book. For my part, it's interesting not only that symmetry is pleasing, but *certain forms* of assymetry as well- asymmetrical elements have to have a "balance".
IMO, the taproot, so to speak, of the phenomenon of the 'pleasingness' quality is survivability. In nature, both smmetry, and *balanced* assymetry, impact the survivability not only of biological organisms, but also of structures, although it's probably easier to consider biological organisms, as there are so many different examples.
For quadrupeds and bipeds, symmetry is important firstly because of its impact upon locomotion, vision, and manipulation of the environment. Picture an herbivore grazing, and the picture it being chased by a predator. For both of them, speed is enhanced by the coordinated movements of their four legs. Although a three-legged creature would be stable (and possibly be able to sleep standing up - which bipeds, as far as I know, cannot do), it's impossible for me, at elat,t o imagine how a three-legged creature could coordinate its limbs so as to reach the speeds of, for example, a gazelle, or a pursuing cheetah, or a horse. Balance makes for efficiency. For humans, walking is a surprisingly efficient means of locomotion, because of the way the human body is (usually) balanced. A thrid arm might sometimes be "handy" (sorry for the bad pun ;) ), but it'd have a negative effect upon balance and therefore upon efficiency of locomotion, which doesnt' make much of a difference in "couch potato" cultures, but does have an impact upon more rigorous lifestyles, esp. the ancient hunting-gathering, and even manual farming, lifestyles of our ancestors. ((Remember that the combine, for example, only was invented and came into wide use in the late 1940's or early 1950's.)) Inefficient locomotion generally makes for slower locomotion, but possibly more importantly, more calorie-intensive locomotion, since it takes more effort - and again, for the vast majority of human history (and, really, even for a gret amhy, if not most, of today's world population), starvation is a constant threat, so anything that made an individual require more calories tended to be dropped from the gene pool, due to the lowered survivability of the individual.
A lack of symmetry affects not only efficiency of locomotion, but also perception. Given that perception is vital for both finding food and avoiding predators, again, in a survivalist/"primitive" mode of existence, anything that negatively impacted it would be unlikely to be passed on to the next generation.
As for "balanced assymetry", it's obvious that the most symmetrical shape would be a sphere (right angles being very uncommon in nature, aside from some rock/crystal faces and formations), and equally obvious is the fact that the closest any biological entity is to a sphere occurs in microorganisms and jellyfish. Higher animals have separated perceptual organs from digestive organs, and both from locomotive organs - IOW, a head, a gut, and legs/fins/wings. However, all of these things also need to operate in the environment of the Earth: gravity, hydrodynamics (whether it be of air or water), temperature, and so on. If the head is too large, the animal wouldn't be able to move forwards, and IIRC the only animal whose main/only direction of locomotion is backwards are squid. Most animals on Earth (and in most of Earth's history) are laterally symmetrical, i.e. symmetrical along the long axis, but assymetrical front-to-back. But that assymetry must be in balance, or else the animal wouldn't be able to move efficiently. Its body, IOW, has to balance all the forces that the animal encounters during th ecourse of its life.
So, while mathematicians talk about 1.618 and 1.414 and other "magic" ratios, it might make more sense to look at animals, and where their "pivit points" are located and which rations permit, or don't permit, them to balance.
Just my own random thoughts ;)
- K.
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Sleeping right behind me is one of my dogs, 15 year old Georgie Girl, who lost a front leg when she was about 6 months old and then last spring had a tumor removed from a back leg that permanently damaged her muscles in that leg. When younger she kept up with the other dogs and caught Frisbees with ease. She still swims well and can walk at a good pace if motivated (meeting new people or dogs). Obviously she is not symmetrical, but locomotion was not seriously disturbed. Her remaining front leg is very heavily muscled. I can envision a 3 legged beast because I have a well functioning one with really 2-1/2 legs. Didn't Larry Niven (sci-fi author) write about that once? EDS
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Permit me to try to sum up the discussion as I see it:

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No, I was talking about efficiency versus inefficiency and the development of symmetry in higher animals and insects and the interplay of that with the fact that most animals prefer mates with a high degree of symmetry. The subject of the original post (and thread) is "WHy is symmetry satisfying", and I was presenting, albeit in summarized form, what is known about the animal kingdom. It's not a matter of "good" or "not good", it's a matter of efficiency and adaptation. If someone dumped a bunch of unwanted greyhounds onto the African savannah, they'd probalby have the speed and agility to outrun lions and even cheetahs. (It's another matter entirely whether they could become successful predators - I've read that feral dogs seldom do to the extent of being able to establish a stable breeding population.) But take a dog who has had a leg amputated, and put it into the same situation, and sorry, but it's goign to be Dinner - being able to "keep up with" oterh dogs in a domestic, meaning protected, situation is completely different from whether three legs is sufficiently efficient at *both* outrunning four-legged predators, *and* hunting prey, so as to survive and establish a breeding population.
The point is that we do *not* live on a planet dominated by trilaterally- symmetrical creatures - there were some, IIRC, very early on in the history of earth, but they disappeared.
I don't understand why the point seems so obscure.
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Starfish, octopus and snake too.

I believe the Marines know how. ;-)

And on the planet of three legged creatures they have no idea what were doing down here. How can you have one leg pushing, one repositioning and another landing when you have four? Either one leg is wasted luggage or you are duplicating motion and there's some critical aspect of locomotion unattended.

The key to walking is lack of balance. Like those jet fighters that are so agile because they are always THIS close to falling out of the sky.

You ain't met my cousin Agnes.

But still, isn't it just satisfying to say "cephalopod". Say it again now "cephalopod." Gastropod. Mmm.... gastropod...

Shouldn't take you much effort to find third party "proof" that those very points fall within the domain of the "interesting" numbers. The truth is out there.
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Snake and octopus are bilaterally symmetrical ;)

Yeah, but they're the exception (exceptionally tough!) ;)

But th epoint would be to match up a three-legged critter with a 4-legged predator. I don't see that the mechanics of the 3-er could be more efficient or even as efficient - which would mean it becomes a meal.

No, biomechanics is an entirely differnt thing. Without continual computer control (which is far faster than human reactions), the things wouldn't get off the ground (or would fall from teh sky). A mroe accurate example is birds - differnt airfoil shapes are adapted to different purposes (c.f. goshawk, falcon, hawk, albatross, and yup, penquins, who 'fly" underwater - air and water, hydrodynamics, the differnce being viscosity/density). Could a three-winged bird fly? Maybe btu the biomechanics would be a nightmare, and nature generally tends to work like a version of Occam's Razor, favoring efficiency over inefficiency.

<L!>
THose are lower animals. I was talking about higher animals, and hwo.why a prefernce for symmetry, or balanaced assymetry, developed. "Preference" implies some sort of higher brain function. Gastropods (IIRC, snails and their ilk) don't, to my recollection, have much in hte way of a brain.

The book referenced by the OP was, IIRC, written about math, by a mathemetician, abotu symmetry. i added "and balanced assymetry".
I have some recollection re: the origin of Phi, but I don't know the origin of 1.414. I've never done an analysis myself of balanced assymetry, because I hadn't previously thought abotu the subject - uit's always been more of an intuitive art-related thing to me. If you know of someone else having done that sort fo study, I'd be interested in the reference, because it'd be, well, interesting to me. As for doing it - yikes, I already have so many other projects going on...and the electronics bits are enough math for me to handle at any one time ;)
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And on the planet of three legged creatures the point would be to match up a four-legged critter with a 3-legged predator. They don't see that the mechanics of the 4-er could be a more efficient or even as efficient - which would mean it becomes a meal.

Yes. Your discussion of computers and eventual side project on three winged birds has no relevance to my point.

Nature tends to work like a human, continually reusing the same solution and rarely thinking outside the box. See "history since the Burgess Shale."

Those are lower animals. I was talking about how fun it is to say the words.

I can't imagine that we prefer symmetry because it helps us run from wolves rather than we prefer symmetry because we are symmetrical. We may be symmetrical because it helps us run from wolves (and quite possibly are not) but our emotions aren't that scholarly.

Square root of two. Has some fun properties.

Don't know if "The Origins Of Architectural Pleasure" gets in to discussion of such, but imagine that it might touch on the symmetry/ quaint issue.

It seems, on quick glance, to be easier to find stuff showing people prefer stuff with "pretty numbers" than that God designed stuff that way. Color me surprised.
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Only if they are inherently more efficient. I'm not an expert in biomechanics, but IIRC, four or two is more efficient than three.

huh? You're the one who brought computer-controlled machines to the question of possible explanations why symmetry is pleasing.

OK.
You're saying "we"/humans, I'm also including other animals. But whatever. It's not even my own theory - I might have come across it in the journal "Nature" but can't recall at this point. Do you have a better/more sensible theory?

As above, I don't know what those are; just haven't looked into it (yet).

Huh?
WHat does that have to do with what you quoted where I said I've too many projects going on to get, at least currently, into a study of animal proportions?
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Bzzzt.
Not "if you recall". If you'll recall, you "can't imagine". There's a big difference there. Are you imagining your argument or are you recalling somebody elses?
It's beside the point. My point is merely that when you say you 'can't imagine' that's pretty much the root of the discussion about symmetry and preference. We like what we know. What we don't know is odd and we "can't imagine" how it might work. My point - on the 3 planet, they have the exact same problem when they look at us. My point - that by limiting ourselves to your imagination we get very suspect answers.

If I understand your statement correctly:
I make no mention of computer controlled machines as a possible explanation for why symmetry is pleasing. I will make no such mention. I do not think I have any belief that such a thing might be true.
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Why do you think it's impossible to do both? (1) From what I recall, X, AND (2) attempts to think of unresearched situations where "not X" is equally viable were unsuccessful in the time I gave to the endevor.

Who can recall the point, with this ever-widening arc of tagents off of other tangents?

No, there *is* data gathered from research showing that, if given a choice, one of the major preferences for a mate is one who is symmetrical (along the linear axis).

Once again, you've massaged what I said and used the incomplete result to claim that I said something which I did not. I can *of course* imagine how three legs might work anatomically biomechanically (duuuh!); but I couldn't think of a way in which a three-legged animal could compete in terms of speed, strength, or agility with a 4-legged predator (or conversely for a predator, prey animal).

And if you went to the planet odf three-legged animals, bring a bunch of 4-legged animals, and released them, would they outcompete the three- legged ones - I think they would, based upon what I recall (yes, RECALL) of anatomy (which, yes, I did study as part of my University coursework), and what I can deduce from it regarding the biomechanics.

Look, you mentioned the jet fighters - which you yourself quote above. And the plain and simple facts are that (1) one cannot consider them without also considering the fact that advanced computer systems are required for tehm to fly, unlike walking, which can be handled (normally at least) by the animal brain; (2) walking is a controlled fall, which isn't the same thing as lacking balance. Anyone can see that I mentioned birds merely because you mentioned jets, because flying is more akin to flying, than it is to walking - IOW, you made an analogy, and I mentioned what is a more apt analogy (flight to flight, rather than flight to walking).
IMO, all you're doing is quibbling, trying to drum up an argument using red herrings drawn across various points, and incomplete quotes carefully selected in an attempt to make your target appear dimwitted.
I'm in the middle of trying to learn some electronics (to make solar- lights for my stained glass scuptures/constructs), and learn what I need to finish several other projects as well, so, although I enjoy real discussion and information exchange, excuse me if I don't have the time or energy to spend on meaningless polemics and drummed-up contention.
- K.
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My point, rather, was that I think you are changing your story not merely telling the parts that suit you at certain times.

Yes, exactly. Look carefully at who is doing the chosing. See below.

Yes, I know. My point is that that is an expectable shortcoming in you and in no way indicative of the truth of the matter.

Yes, I know you think so. And _I_ think you may well be wrong.

Your coursework in biomechanics should have left you well able to understand what I meant.

Yes, yes I did. So what?

The plain and simple facts are that you can not read.

I'll leave you and somebody else to argue the compare/contrast between "advanced computer systems" and the apparently sluggardly backward incompetent human brain.
Irrelevant. I told you that in the last post and yet you keep arguing about it. I see your point. I argee. And it is utterly irrelevant to what I said. What I meant. What I was illustrating. Yet you fall all over it even after being told that it is not in the least bit relevant and you are barking up your own tree. I can not make your life any easier for you if you insist on making it so hard.
The interesting point you seem to have missed about the unstable aircraft is that w/o the control systems they fail. Same too humans.

See, you can read! One sentence out of pages of irrelevance and wrong accusatory snideness (see below). Thanks be to Don.
Well phrased. You're right. We don't walk because we lack balance, we push ourselves "off balance" and use our balance to recover.
Let's think about the planes... they are being pushed by engines and held up by wings. People are being pushed by legs and held up by long stiff bones (the mechanics of the two are different by far, but I think the analogy should suffice for our purposes. We'll see soon enough). A paper airplane, pushed, will tend to glide forward, nose first in a pleasant manner. A person, pushed, will tend to fall over. An X-29, pushed, being unstable, will tend to turn around and do all sorts of ugly things for a plane. A person, pushed, being unstable, will tend to fall over and do all sorts of ugly things for a person.
The behaviour seems the same, but

The _key_ to walking is not lack of balance, you are right. The X-29 relies on instability to increase agility. The human may use a moment of instability as a transition point (at "best" here) as a stage in the system of walking, but no, the analogy I was thinking of just doesn't cut it. Sorry.
Of course, that doesn't mean three legged beasts couldn't kick four legged butt.

Yes, I could see that. And it was a very poor reason. It underscored that I didn't manage to convey my point to you in the least. Perhaps because it was such a poor analogy (mine) you found yourself grasping at straws to answer it. A simple "I'm sorry, I don't follow." or today's nice summary of what walking is would have been a better tactic.

It was more apt to what you were thinking. Incredibly unrelated to what I was trying to say.

No, not at all. I'm trying to make sure you know that I am not saying, trying to say, implying or even thinking about the many things you ascribe to me when you talk to yourself in reply to my tiny little observations. That, and I was trying to get you to see that your argument about the inferiority of three legged beasts and their lack of bilateral symmetry was baseless bigotry born of "ethnocentrism" and lack of imagination. AND that this bigotry ties in exactly with the human preference for symmetry. I repeat, did you ever address? - we like what we are. I add - that's sort of a survival trait right there. I add - and we fear that which we are not.

That's pretty Internet 101 of you.
Regarding "real discussion," I was trying very hard to keep to real discussion of those things I was discussing. You can't have enjoyed it because you kept trying to drag me off into places of your own concern. I told you "I'm not talking about that and I don't care." Never once did you say "oh, oops. What are you talking about?" No, rather, you kept (no doubt (and in contradiction to your above protestation), keep) going on about shit I don't care about in this context, shit I'm not talking about in this context, and long tedious paragraphs responding to... stuff I never said, in ways that make it abundantly clear you aren't understanding the most simple of my statements and aren't interested in so doing.
So, say "whatever," and pretend you win because you've got IMPORTANT AND REAL things to do.
I know that I, for one, have clearly "lost".
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An expected shortcoming? Or your own misinterpretation?

Fine.
Flight is a more of a matter of mechanics interacting with fluid dynamics than is walking, or even running - air resistance is a crucial factor in flight, but insignificant in walking, and not very significant in running.

As above. The analogy was a thin comparison.

Right, I can't read.

I know. I made the error of trying to get back to the main point, which was mentioning some theories (the impication being evolutionary theories, tho' I supposed I ought to have stated that more obviously)of why animals (including humans) might prefer certain physical arrangements.

Right.
Which was why I brought in birds, which you claimed to be irrelevant, but is not. The point was about "why symmetry", and more specifically, What is efficient about bilateral symmetry. Symmetry makes for a smoother, more efficient interaction with air (expressed using similar math to that which is used to study underwater movement).

I wasn't "grasping at straws", I simply brought in the more appropriate analogy.

Tactic? So, uh, what, you thought I was picking on you or something?

I don't talk to myself more than anyone else, including you.

No, I sketched (i.e. physically, with a pencil on paper) some possible general skeletal configurations - excuse me is that is "unimaginitive" - and tried to imagine how the musculature would work. And I remained unconvinced that such an anatomy would be efficient enough to compete with animals exhibiting bilateral symmetry.
My statements were based upon what I know about anatomy and of how animals move through their environments.
The only "baseless bigotry" seems, sorry, to be your own.

I wasn't merely talking about humans. The question of "stranger fear" isn't the same as why there seems to be an instinct to consider symmetry beautiful, an instinct with is observed in various animals, at least in part, when oen studies mating preferences.

Well, if that's how you want to interpret it.

I wasn't "dragging you off" to anywhere, I was trying to keep to the question raised by the OP.
And, just a question, but, if you don't care what someone is talking about, why shoudl they care what you're talking about?

Well, if that's nto the pot calling the kettle black, I dunno what is.

I didn't say "important and real", I said "alot". They're important to me but obviously not inportant to everyone else. As for "pretend", well, if the best you can do is call me a liar, that's not even worth a response.

You've lost whatever respect I used to have for you. Which obviously means nothing to you anyway, although I'm glad, in a way, that I found out your true opinion of me - it will make for better time-management for both of us.
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Amazing.
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Picture a pentagram (going with the most common number of appendages) - technically, yes, you can "split it down the middle" - one point is halved, and two whole points are on either side.
Starfish (which are invertebrates) mating choices are based upon chemoreceptors - although they do have an eyespot at the end of each arm, evidence indicates that these eyespots only sense general dark and light, not acute enough to use visual cues for mate-selection. I think it can also be said that they exhibit pentaradial symmetry. (Just to be sure, I checked Google, and that info is readily available.)
I did some Googling - the following sites, which deal with vertebrate locomotion, may not be of interest to you, but if they are, enjoy ;) : http://www.efeld.com/evolution / http://www.earthlife.net/mammals/locomotion.html http://icb.oxfordjournals.org/cgi/reprint/42/1/158 - "Stability and Manoeuvrability of Terrestrial Vertebrates" http://www.cs.bham.ac.uk/~txm/thesis.pdf - "Evolution, Complexity and Progress in Natural and Artificial Systems" (long, 213 pages, I haven't yet finished reading it)
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For my money, no. Yes, you can cut a starfish, either even or odd legged, along one axis and say "see, mirror images" but there is no single axis of that symmetry. You can cut here OR here and get the same answer. So we call them radially symmetrical.
And when I was a kid the plain starfish we had had one "eye" that was asymmetrically placed. But then, I've only got one asymmetric liver too, so maybe that arrangement of guts and stuff doesn't counts as much.
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THis is a short but decent bit on bilateral symmetry in animals: http://evolution.berkeley.edu/evolibrary/article//arthropods_04 and where would we be without the required Wikipedia reference <g!>: http://en.wikipedia.org/wiki/Symmetry_(biology)
What they don't seem to address is that, although *technically* a 5-armed starfish could be visually 'sliced' such that two approximately-mirrored halves result, the major factor seems to be that there is a clear "front"/head-area/direction-of-forward-movement (or maybe all of those three?) creating a "head to butt" line.
So, for a mathematic/gemoetrical example, you could have a star-shaped window which is divided into two mirror/equal halves by a corner or brace or whatever. But with living organisms, the "head to butt" line (the spine) seems to define bilateral symmetry, and the lack there of in starfish (and other enchnoderms, and polyps, for example) seems to define tham as not being bilaterally symmetrical, but radially symmetrical.
In a building, I think that having equually-sized windows, or at least a limited size-range, "feels" more ordered and more balanced - most livng organisms tend to have a size-range for their parts, and it's possible that a preference for that srt of order could be, if not directly instinctual, then perhaps an inherent preference for visual order; if learning in not just a matter of acquiring facts but also of organizing/ordering/calssifying those facts to aid in retention and understanding, it wouldn't surprise me if that 'spilled over' so to speak into the visual realm.
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wrote:

Actually "classic" window sizing usually makes the windows of the main floor larger (usually longer) that windows of subservient floors. This identifies the "important" areas and pleases the eye. The dimensional change need not be large, just enough to percieve. Often the difference between a beloved building and an awkward building lies in very subtle relationships between elements and how the eye is pleased or not pleased by them. Remember that "God is in the details" and "If architects built buildings without engineers they would fall down, but if engineers built buildings without architects they would be torn down."

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wrote:

True. I think that's part of what I was calling (in quotes as it wasn't/isn't, AFAIK, a technical term) "balanced assymetry". It's somehting you see mose of the time in nature - looking at feathers, scales, and leaves, tehre is a variation in size between teh individual things, usually related to their position on the creature/plant, also usually gfalling within a certain overall range. IOW, the architectural elements, similarly to the natural elements, vary in an ordered manner, not just "willy-nilly".
I was thinking about 1.618 and 1.414, .75, and a few others that seem to recur in human-made items, and why they do recur, and they seem to echo common natural ratios.
I don't think it's only a matter of "visual habit" for lack of a more technical term; starting at the atomic level, moving through the reflection of of atomic and molecular shapes in crystals large enough to be visible to the human eye, and on to the structures of plants and animals, there is a perceivable order to the structure of things, and it's been theorized that this order is sensed at a fundamental level by the brain, even primitive brains, and was carried through to the human brain. The brain seeks to impose order even upon random disorder, which psychologists theorize is the reason that children "see a bunny in that cloud", and why other shapes/images seem to occur in random patterns.
I don't say that visual habits don't exist, but that there is some sort of fundamental (and subconscious) preference for order, and that ratios such as 1.414 were (and are) part of the human visual vocabulary because they do coincide with the organization of natural structures.

And I think it's usually subconscious. I don't know whether any study has ever been done that takes pictures of things which are considered beautiful, and very subtly alters the proportions/composition/etc., then asks people to judge which version is the "most beautiful" - I'd be interested toknow whetehr one has been done, and if one hasn't, I think it'd be interesting to do such a study.
Anyhoo, I discoverd what you mentioned when I'd started modeling builidngs in 3D. As you said, the difference (between, for ex., upper storey windows and main-storey windows) is pretty small, but i's true that, when I first modeled something and (erroneously) made all teh windows identical, it was definitely less pleasing as a model than it was when I fixed the error (after measuring the windows).

True, and in so many things ;)

I never hear that quote, but it's a good one to remember ;)
Fascinating phenomenon.
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wrote:

The next system that is used when symmetry isn't is eccentricity. The CG is located with the building mass being used. The main entrance can be placed at the mass CG for balance around the entrance. Of course the rest of design properties like proportion, repetition and other properties of design etc. needs to be considered. this system can be used for beautiful buildings if done correctly.
CID...
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