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I have two small children, and so I wind up watching a lot of children’s cartoons. A geometrical concept that leaps out at anyone who sees any significant amount of children’s cartoons is the geometric support for differentiating between the good guys and the bad guys, namely, that the good guys are highly rounded in appearance, and the bad guys are highly spiky in appearance. A spot-on example of the former is Mickey Mouse, and a spot-on example of the latter is a witch (pointed hat, long pointed nose, long menacing fingers, broomstick). In passing let’s note the aural correlation: a (more or less) mellifluous voice versus a high-pitched crackling menacing laugh.

So, here’s the question: Has anyone ever done a mathematical, or quantitative, analysis of this phenomenon?

It seems like the notion of convexity is right in the middle. What is round is convex, but what is convex can be pointed. Now, the notion of convexity has been investigated deeply and widely, has it not? I would be greatly surprised if there has not been developed a notion of convexity relative to some threshold, which could be, or perhaps has been, used to define the notion of roundedness precisely (eg something as simple as “the radius of curvature at every point is at least h”, where h is a fixed positive value characteristic of the environment, or venue).

Has this phenomenon perhaps even been named, perhaps within the field of Psychology? If not, can someone think of a good name for it?

It seems that the proposition friendly=rounded (equivalently, hostile=pointed) is pretty much instinctual. (How many times does a child get told by an elder, “Don’t point. It’s not polite.”? And of course there’s the pointing of swords and guns at the enemy. Or again, contrast the friendliness of the blue sky with the menace of a lightening bolt from an overcast sky.)

Although this phenomenon is rampant in the domain of children, it seems to be exploited, consciously or unconsciously, in advertising. There is a reproduction of an old poster on display on the back bulletin board of one of the classrooms I currently teach in that illustrates this. It is an advertising poster for a brewery company. It shows a woman sitting in a wicker chair. She is turned, facing the viewer. Her features are very rounded, including the curl prominently displayed in the middle of her forehead. Her fingers, a possible indicator of hostility, because pointed, if not handled properly, are dealt with as follows: In both cases the fingers are pointing orthogonally to the line of sight of the viewer, and her chin is resting on the back of her right hand, depressing the back of the hand, clearly rendering that hand ineffective as any kind of threat.

Regarding the identification of this poster, I can also offer the following: Beneath the picture of the woman, on each side, is a picture of a bottle of the product being advertized. Among the linguistic symbols above the picture is the following text:

AKTIESELSKABET UNION BRYGGERI

SCANDINAVIAN BREWERY CO., LTD

So, perhaps a great thesis topic for a graduate student (perhaps in Mathematics proper, or in “Quantitative Marketing Research”) would be to give a quantitative report of the role that “roundedness as friendliness” plays in advertizing, or elsewhere (if, indeed, this hasn't already been done, which gets back to my question).

Regards, Mike Jones American expatriate in Beijing 8.May.2011

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You came up with all of this just by watching cartoons :O...? –  Mixxiphoid May 7 '11 at 18:31
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This is more psychological than mathematical, I reckon. I seem to recall a study that characters geared towards children are built to have as few "corners" as possible. –  J. M. May 7 '11 at 18:38
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Quite interesting observations. I think your allusions to knife , like nails, swords, etc., can easily provoke a sense of danger (which might easily generalize to "pointed/sharp"...with respect to "round", there may be a couple of confounding factors: especially when an image invokes perceptions of both roundness and softness. Your question has me wondering if some of these images, and the emotional attraction/avoidance like/dislike responses that they invoke might not have, at some degree, an adaptive, evolutionary origin. –  amWhy May 7 '11 at 18:42
    
From a mathematical point of view, it would be feasible I think to create simple numerical criteria for "roundedness". But the most difficult thing I think would be to truly isolate this parameter from many other psychological parameters. Mickey also has a funny voice for example. –  Joel Cohen May 7 '11 at 19:08
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Closed. This seems much more like a psychological than a mathematical question to me. (That doesn't mean there's no mathematical content, but it does mean that you should ask psychologists about it, not us.) –  Qiaochu Yuan May 7 '11 at 19:36
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closed as off topic by J. M., Sivaram, Zev Chonoles, Qiaochu Yuan May 7 '11 at 19:36

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