We know from IQ tests that working memory abilities vary. Those with aphantasia can’t visualize at all, while others report that not only can they visualize a tiger, they can count its stripes. My visualizations are not that stable. The number of stripes would probably change as I attempt to count them. But visual thinking can be improved with practice, at least in my own experience. Things that took a lot of effort to visualize the first time become simple recall after that. The bigger your bag of tricks, the more likely you can find one that applies to a novel situation.
Visualizations need not be static images. They can have motion as well. I can rotate simple 3-D shapes in my mind, for example. Rotating a cube is pretty easy. I can even do an icosahedron, though that one took some practice. But counting the leaves on a tree would be too difficult, never mind rotating the tree without changing (or glossing over) their number. There are limits to the resolution. You can also do transformations other than rotations, like scales, shears, extrusions, etc. These visualizations are useful in computer graphics and in topology.
In the case of mathematics, I find visualization most useful for generating examples, especially counterexamples. Using the visual query process I described, one can try to query for a shape that meets certain constraints. Sometimes one example (or counterexample) is all it takes to prove a theorem. Sometimes the query produces the example, but sometimes it fails to meet all the constraints and I have to query that part again. Pointing out the part that failed a constraint can bring more examples to mind. You have to give these mathematical objects a visual form to gain the benefits of visual thinking, but there are many morphisms one might try. Besides single examples, you might also be able to enumerate a set of them, or notice a pattern that can be repeated to infinity.
I can generate candidate visualizations much faster in my head than I can draw them on a whiteboard, but then communicating that insight to another person may require a diagram.
A visual thinking riddle: go in one hole and come out three. What am I? I solved this one visually pretty quickly. Try to generate candidate visualizations and see if you recognize the shape.
(Answer: grrfuveg.)
Do you visualize the icosahedron as one object or do you split it up and consider each separately, but reminding oneself that it is actually one object?
My answer to your visual thinking riddle is: breath in through your mouth and breath out through mouth + nostrils. But I can’t decipher your anagram!
Do you visualize the icosahedron as one object or do you split it up and consider each separately, but reminding oneself that it is actually one object?
I have looked at a d20 long enough and from enough angles (it’s very symmetrical) to have memorized the whole icosahedron, and can visualize it that way, at least as an opaque object from the outside.
But the mnemonic technique of chunking is a valid strategy for visualization. Short-term memories must be “refreshed” or they fade away, but if you juggle too many at once, you’ll drop one before you can get back to it. Making each face a chunk would be 20, which is too many. 3-5 chunks is a more reasonable number. My favored decomposition of the icosahedron is into a pentagonal antiprism with pentagonal pyramid caps. That’s 3 chunks, and two of them are the same thing. Other decompositions may be useful depending on what you are trying to do.
More complex objects can be visualized as hierarchical decompositions, though not always in their entirety. Recognition is not the same as recall. The resolution of a weak visual memory may be just enough to recognize a new example (but too low to count the faces, say). A really low resolution image is more of a handle than a structure, but it can point you to the memory of the real thing.
We know from IQ tests that working memory abilities vary. Those with aphantasia can’t visualize at all, while others report that not only can they visualize a tiger, they can count its stripes. My visualizations are not that stable. The number of stripes would probably change as I attempt to count them. But visual thinking can be improved with practice, at least in my own experience. Things that took a lot of effort to visualize the first time become simple recall after that. The bigger your bag of tricks, the more likely you can find one that applies to a novel situation.
Visualizations need not be static images. They can have motion as well. I can rotate simple 3-D shapes in my mind, for example. Rotating a cube is pretty easy. I can even do an icosahedron, though that one took some practice. But counting the leaves on a tree would be too difficult, never mind rotating the tree without changing (or glossing over) their number. There are limits to the resolution. You can also do transformations other than rotations, like scales, shears, extrusions, etc. These visualizations are useful in computer graphics and in topology.
In the case of mathematics, I find visualization most useful for generating examples, especially counterexamples. Using the visual query process I described, one can try to query for a shape that meets certain constraints. Sometimes one example (or counterexample) is all it takes to prove a theorem. Sometimes the query produces the example, but sometimes it fails to meet all the constraints and I have to query that part again. Pointing out the part that failed a constraint can bring more examples to mind. You have to give these mathematical objects a visual form to gain the benefits of visual thinking, but there are many morphisms one might try. Besides single examples, you might also be able to enumerate a set of them, or notice a pattern that can be repeated to infinity.
I can generate candidate visualizations much faster in my head than I can draw them on a whiteboard, but then communicating that insight to another person may require a diagram.
A visual thinking riddle: go in one hole and come out three. What am I? I solved this one visually pretty quickly. Try to generate candidate visualizations and see if you recognize the shape.
(Answer: grrfuveg.)
Do you visualize the icosahedron as one object or do you split it up and consider each separately, but reminding oneself that it is actually one object?
My answer to your visual thinking riddle is: breath in through your mouth and breath out through mouth + nostrils. But I can’t decipher your anagram!
I have looked at a d20 long enough and from enough angles (it’s very symmetrical) to have memorized the whole icosahedron, and can visualize it that way, at least as an opaque object from the outside.
But the mnemonic technique of chunking is a valid strategy for visualization. Short-term memories must be “refreshed” or they fade away, but if you juggle too many at once, you’ll drop one before you can get back to it. Making each face a chunk would be 20, which is too many. 3-5 chunks is a more reasonable number. My favored decomposition of the icosahedron is into a pentagonal antiprism with pentagonal pyramid caps. That’s 3 chunks, and two of them are the same thing. Other decompositions may be useful depending on what you are trying to do.
More complex objects can be visualized as hierarchical decompositions, though not always in their entirety. Recognition is not the same as recall. The resolution of a weak visual memory may be just enough to recognize a new example (but too low to count the faces, say). A really low resolution image is more of a handle than a structure, but it can point you to the memory of the real thing.
That’s because it isn’t an anagram. ROT13 :)