Why do so many zooms of the Mandelbrot set M focus on just one portion of the seahorse valley? What is more demanding of human attention in that region than others?

John Derbyshire talks about it on this page. And Niall Ryan has a zoom about it on this page

Something very funky is going on here. Human aesthetic senses are good at distinguishing things that are especially pleasing to the eye. Why are bulbs around here more pleasing than others? What is special about this one?

(from wikipedia):

And here’s a zoom I found on youtube.

And from the revered Heinz-Otto Peitgen:

Something is tickling our aesthetic senses. We need to figure out what.

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2 responses to “There is a distinguished place in the Seahorse valley.”

The ideal diet for a human being has a fair amount of variety. Crunchy, smooth, sweet sour, the fact that we get bored with food that is always the same is a matter of survival. Our preference for variety ensures that we have all of the nutrients we need, variety also makes us more robust. (So what if there are no fish this year, we’ll eat clover and berries, or hunt rabbits etc. ) In the same way to support these needs we are attracted to environments with variety. A river, a lake, a water fall, some rolling hills and some mountains, but also some planes, a little stand of forest trees, such places are considered beautiful and painted as landscapes since, I think, at some deeper level we know we can find the variety in food and resources required for survival.

The beautiful fractal may have the same kind of variety, several different kinds of self similarity, operating on different scales, sweeping diagonals and curves.

Of course rigid order can also be attractive. I view this as the order we hope to impose on the world around us.

But in nature, and I would include these mathematical sets in nature, the beauty comes from the borders between environments where the diversity the support human life is most often found.

The only caution I was make here to any visiting mathematicians is that we must not be too captivated by the beauty: the desire for rapidity of mathematical analysis demands that we not let ourselves be too carried away by detail: for the discovery of generalities requires a willingness to be flexible with what one means by “rigid order”: we quest for the abstractions which are most provident in terms of understanding: it is good to be captivated, but only for a little while.

For instance, somewhere very far elsewhy: imagine Gerestheo: he is obsessed by angles. He has printed a two hundred page tome that has angles sorted by spikiness. He gives them descriptive names. He catalogues where they occur in the natural world. Gerestheo is sold on acute angles, but he doesn’t call them that. He is deeply suspicious of obtuse angles, and plain doesn’t believe in right angles. Sparanonquil comes by and shows him the sine and the cosine, and how angles and and trigonometry. That doesn’t make Gerestheo’s descriptive classification any less valid, but it does point out the dangers of Pokemon mathematics: “you’ve got to catch them all”. We need to be concerned that the systems of names for things we impose on the objects of ours study do not inhibit our ability to abstract: mathematics is not about the arbitrary names that we choose for those objects of our study. In the microscopic, what name to we give a particular variable (so that it does not conflict with other variables), in the macroscopic, making sure our system for classifying and determining the properties of new, bizarre objects is good enough not to let us be bewildered by them. Let us not fear the forest.

The ideal diet for a human being has a fair amount of variety. Crunchy, smooth, sweet sour, the fact that we get bored with food that is always the same is a matter of survival. Our preference for variety ensures that we have all of the nutrients we need, variety also makes us more robust. (So what if there are no fish this year, we’ll eat clover and berries, or hunt rabbits etc. ) In the same way to support these needs we are attracted to environments with variety. A river, a lake, a water fall, some rolling hills and some mountains, but also some planes, a little stand of forest trees, such places are considered beautiful and painted as landscapes since, I think, at some deeper level we know we can find the variety in food and resources required for survival.

The beautiful fractal may have the same kind of variety, several different kinds of self similarity, operating on different scales, sweeping diagonals and curves.

Of course rigid order can also be attractive. I view this as the order we hope to impose on the world around us.

But in nature, and I would include these mathematical sets in nature, the beauty comes from the borders between environments where the diversity the support human life is most often found.

The only caution I was make here to any visiting mathematicians is that we must not be too captivated by the beauty: the desire for rapidity of mathematical analysis demands that we not let ourselves be too carried away by detail: for the discovery of generalities requires a willingness to be flexible with what one means by “rigid order”: we quest for the abstractions which are most provident in terms of understanding: it is good to be captivated, but only for a little while.

For instance, somewhere very far elsewhy: imagine Gerestheo: he is obsessed by angles. He has printed a two hundred page tome that has angles sorted by spikiness. He gives them descriptive names. He catalogues where they occur in the natural world. Gerestheo is sold on acute angles, but he doesn’t call them that. He is deeply suspicious of obtuse angles, and plain doesn’t believe in right angles. Sparanonquil comes by and shows him the sine and the cosine, and how angles and and trigonometry. That doesn’t make Gerestheo’s descriptive classification any less valid, but it does point out the dangers of Pokemon mathematics: “you’ve got to catch them all”. We need to be concerned that the systems of names for things we impose on the objects of ours study do not inhibit our ability to abstract: mathematics is not about the arbitrary names that we choose for those objects of our study. In the microscopic, what name to we give a particular variable (so that it does not conflict with other variables), in the macroscopic, making sure our system for classifying and determining the properties of new, bizarre objects is good enough not to let us be bewildered by them. Let us not fear the forest.