Can you solve problems in geometry in your head? For example: How many edges does a dodecahedron have? Prove that the medians of a triangle are concurrent. How long is the altitude of a regular tetrahedron? How many Platonic solids are there in 4 dimensions?
It might be worth attempting to see how you perform on certain types of spatial thinking problems that most people claim to use imagery to solve (although no correlation seems to exist between spatial thinking ability and the vividness that people report their imagery to have). Try to solve the problem below, in your head, without drawing diagrams or making calculations on paper or anything like that.
The four narrow sides of a 1 cm by 4 cm by 4 cm block are painted red. The top and bottom are painted blue. The block is then cut into sixteen 1 cm cubes.
How many cubes have both red and blue faces?
How many cubes have one red and two blue faces?
How many cubes have no painted faces?
Most people say they use imagery to do this, and count the relevant cubes in their image. Were you able to solve all or any part of the problem at all? Did it seem very difficult? How, in fact, did you solve it (if you did)? Did you have to consciously employ any formal knowledge of geometry or other mathematics (beyond counting)?
When I solved this, I had the interesting experience of Imagining the 4x4 block of 16 blocks, noting that the outside ones (all but 4) had red paint on them, and all of them had blue paint… but I only “put” blue paint on the top. My diagram was flat, oriented like a pancake. None of this was Mental Imagery. Then when I was asked how many cubes had red and blue faces, I felt around the edges of the block. Motor/haptic mental imagery. Then when I was asked how many cubes had 1 red and 2 blue faces, I immediately thought the question was 1 red and 1 blue since I didn’t have blue paint on the bottom in my model (I’m not sure if I had a bottom in my model). I thought “when would they have more than 1 red? ah the corners”, and then had the distinct vivid motor mental imagery of moving my hand and touching two non-corner side blocks on the left of my model, then two at the far side, then two on the right, then two on the near side, counting “2, 4, 6, 8”. This was a different experience than my usual Imagining… but I’m not sure if it was qualitatively different or just more “vivid” motor mental imagery.
Did anyone else have trouble recalling the red vs blue sides? (based on my experience with this (below), it seems as though my mental association was essentially “top and bottom are same” and “narrows are same” but neither really had a color. When I close my eyes, I don’t see “red” or “blue”)
At first I was imagining a 1cm by 1cm by 4cm block. I then realized that getting the 16 cubes of 1cm each out of this wasn’t possible and then went to the accurate idea of a 4 by 4 by 1. I realize I am having a great deal of trouble going from a 1cm by 4cm rectangle and then adding depth, while the 4 x 4 square I can add the 1cm of depth much easier. I can rotate the image of the 4x4x1 around in my head and yet cannot do the same with the 4x1x4 (despite the fact that I recognize that they are the same image).
Adding colors: four narrow are red, two fat are blue. Got it. Break it up into cubes.
Red and blue faces requires it touch the outside of the 4x4 square. 4 along top side, four along bottom side, and two each on the left and right (already counted the corners).=12 total red and blues.
One red and two blue… which ones were red and which were blue again? I know that the four narrows are the same and that the top+bottom are different… and just logicked that if the question is one red and two blue, that means the narrows were blue (the reds don’t touch->top and bottom). To be two red and one blue… WAIT—that logicking doesn’t work because the edges have Top side bottom. So I’ll look back at which sides were which color. In my mind I just have “top and bottom are same color” but that’s not assigned to red or to blue. Okay—top and bottom are blue. This means along the edge of the 4x4. uh… the perimeter again. So 4+3+3+2=12. Unless you’re asking for two red and exactly one blue, in which case it’s the corners… wait—two red? red was the narrow side color. i think i switched them again. Final answer: to have two red and one blue you must have two red, meaning the narrow side meets a narrow side, which only happens in four places. Four.
Just re-read the question. You asked for one red and two blue. All of them except the middle 4 blocks, so 12.
No painted faces? All of them are painted on at least one side: we painted three sides of the block, each of the 16 cubes we chopped the block into touched at least one side. 0 are completely uncolored.
This is very interesting. I am having trouble understanding the experience of imagining the 4x4 block of 16 blocks well enough to note that there are four interior blocks w/o red paint on them without picturing them.
I could imagine that this could be done with just logic (reasoning about how many blocks there must be in different categories, which is maybe how I would do it if the problem were more complex, or took place in four dimensions for example), but you said you had a diagram...
So it sounds like you did have mental imagery, it was just 2-d instead of 3-d.
But apparently that wasn’t very vivid, because you still had to do the haptic imagery thing. How vivid is the experience of this motor mental imagery for you? I’m wondering if I’m missing out on that in the way that you’re missing out on more vivid visual mental imagery.
Summary: Thinking about geometry problems gives me access to visual mental imagery of lines, sometimes pretty stable and controllable lines. A few at a time.
Dodecahedron, stream of thought: 20 sides, right? What’s the… oh you asked edges. 20, clearly. Wait a dodecahedron is 3D. Okay I get the question now. Um, it has 20 sides… they each join or rather each edge joins a pair… aren’t they pentagons? So each side, face I guess, has 5 edges, that’s 20x5 is a hundred edges counted twice for 50 edges. --- During this time I Imagined a vague ball-like thing and Imagined the face of it nearest me and knew that it was a pentagon. Now I’m going to look up the answer before commenting, but I promise I’ll leave all of this even if I’m wrong. Oh it has 12 faces, oops. Then it’s 12x5/2=30 edges instead.
Medians: I Imagine an isoceles triangle and Mental Image the two equal sides of that triangle. I Imagine the vertical median and Mental Image a dark vertical bar across my visual field (this is with eyes closed btw). The bar quickly morphs into a jagged dark lightning bolt thing and back again and things turn amorphous. Um, I don’t know why the medians would be concurrent. Medians split the side, right? So… base times height, they split the area in two… base times height over two rather, whatevs. Okay. Do I know anything else about medians? No… let’s have two medians, clearly they intersect somewhere. I Imagine a scalene triangle with its base horizontal, as on a whiteboard in front of me, Mental Imaging the base as a thick bright line and the two other sides as deformations in the static. I mean it looks like the medians come kinda close. I want to draw things though, I’m losing track and it would take a lot of effort to prove this in my head.
Altitude: Imagining a pyramid with a dotted line from the tip to the middle of a triangular base. Mental Image is the lines of a triangle base with three vertical lines coming up like it’s a triangular prism, they won’t go together to make a point but hey 6 lines is a lot at once, neat. Side length s says triangle altitude is sqrt(s^2-s^2/4)=(sqrt(3)/2) s. I… am not sure where the triangle comes from that I can get the tetrahedron’s altitutde from, though. I want to draw. I Imagine the dotted line and try to make a triangle but I have to explicitly check “what is this line?” rather than seeing it. That’s like part of a triangle altitutde… oh hey the base triangle, I got a symmetric three interior lines both Imagined and Mental Imaged and there are some isoceles triangles there. A needed unknown x, x again, and s. And clearly it’s 30/30/120. So also it’s 30/60/90 with x, (sqrt(3)/2) s-x, and s/2. So x is the hypotenuse and is 2/sqrt(3) times s/2. x is s/sqrt(3). I’ve got (and I’m Imagining, and my Mental Image is like the Imagining except hella distorted but hey it’s there!) then a triangle with one edge the edge of a face s, one the altitude, and one s/sqrt(3). The altitude then is sqrt(s^2-s^2/3)=sqrt(2/3) * s. Checking… yep.
Platonic solids: Um. Four dimensions, huh? I Imagine a cube. Now it’s stretched and I Mental Image the static elongating, which lasts for not long. A hypercube must work, right? What is a four dimensional Platonic solid. It’s a 4D thing with regular 3D things as “faces”? Okay… how the hell does that work. If I can take a 3D thing, morph it over time until it’s back to a 3D thing, and interpret those morphs as… the same 3D thing? That doesn’t make sense, the morphs will be 1D. I am confused. I will look up four dimensional Platonic solids now. Okay, confusing.
This is great. More stream of consciousness while Guy solves math problems please.
I thought it was interesting that it was easier for me to picture the proper shapes than it was for you (I had no trouble getting the lines of my pyramid to join together, and I could easily imagine where the line for the altitude of the tetrahedron went), but you thought of the relations between line segment lengths and came up with the formulas for them much more quickly than I would have.
One thing I want to clarify though, when you said you were imagining the pyramid and dotted line, and then your mental imagine didn’t match that correctly—were you first successfully imagining the pyramid and dotted line, and then trying to also have a mental image, or when you said you were imagining did you just mean that you were starting to form the mental image? And if the former, what did this imagining consist of, other than just awareness of the abstract idea that an altitude should go from a face to its opposing point?
Can you solve problems in geometry in your head? For example: How many edges does a dodecahedron have? Prove that the medians of a triangle are concurrent. How long is the altitude of a regular tetrahedron? How many Platonic solids are there in 4 dimensions?
I was given an excellent geometry problem by Dr. Nigel Thomas.
When I solved this, I had the interesting experience of Imagining the 4x4 block of 16 blocks, noting that the outside ones (all but 4) had red paint on them, and all of them had blue paint… but I only “put” blue paint on the top. My diagram was flat, oriented like a pancake. None of this was Mental Imagery. Then when I was asked how many cubes had red and blue faces, I felt around the edges of the block. Motor/haptic mental imagery. Then when I was asked how many cubes had 1 red and 2 blue faces, I immediately thought the question was 1 red and 1 blue since I didn’t have blue paint on the bottom in my model (I’m not sure if I had a bottom in my model). I thought “when would they have more than 1 red? ah the corners”, and then had the distinct vivid motor mental imagery of moving my hand and touching two non-corner side blocks on the left of my model, then two at the far side, then two on the right, then two on the near side, counting “2, 4, 6, 8”. This was a different experience than my usual Imagining… but I’m not sure if it was qualitatively different or just more “vivid” motor mental imagery.
Did anyone else have trouble recalling the red vs blue sides? (based on my experience with this (below), it seems as though my mental association was essentially “top and bottom are same” and “narrows are same” but neither really had a color. When I close my eyes, I don’t see “red” or “blue”)
At first I was imagining a 1cm by 1cm by 4cm block. I then realized that getting the 16 cubes of 1cm each out of this wasn’t possible and then went to the accurate idea of a 4 by 4 by 1. I realize I am having a great deal of trouble going from a 1cm by 4cm rectangle and then adding depth, while the 4 x 4 square I can add the 1cm of depth much easier. I can rotate the image of the 4x4x1 around in my head and yet cannot do the same with the 4x1x4 (despite the fact that I recognize that they are the same image).
Adding colors: four narrow are red, two fat are blue. Got it. Break it up into cubes.
Red and blue faces requires it touch the outside of the 4x4 square. 4 along top side, four along bottom side, and two each on the left and right (already counted the corners).=12 total red and blues.
One red and two blue… which ones were red and which were blue again? I know that the four narrows are the same and that the top+bottom are different… and just logicked that if the question is one red and two blue, that means the narrows were blue (the reds don’t touch->top and bottom). To be two red and one blue… WAIT—that logicking doesn’t work because the edges have Top side bottom. So I’ll look back at which sides were which color. In my mind I just have “top and bottom are same color” but that’s not assigned to red or to blue. Okay—top and bottom are blue. This means along the edge of the 4x4. uh… the perimeter again. So 4+3+3+2=12. Unless you’re asking for two red and exactly one blue, in which case it’s the corners… wait—two red? red was the narrow side color. i think i switched them again. Final answer: to have two red and one blue you must have two red, meaning the narrow side meets a narrow side, which only happens in four places. Four.
Just re-read the question. You asked for one red and two blue. All of them except the middle 4 blocks, so 12.
No painted faces? All of them are painted on at least one side: we painted three sides of the block, each of the 16 cubes we chopped the block into touched at least one side. 0 are completely uncolored.
This is very interesting. I am having trouble understanding the experience of imagining the 4x4 block of 16 blocks well enough to note that there are four interior blocks w/o red paint on them without picturing them.
I could imagine that this could be done with just logic (reasoning about how many blocks there must be in different categories, which is maybe how I would do it if the problem were more complex, or took place in four dimensions for example), but you said you had a diagram...
So it sounds like you did have mental imagery, it was just 2-d instead of 3-d.
But apparently that wasn’t very vivid, because you still had to do the haptic imagery thing. How vivid is the experience of this motor mental imagery for you? I’m wondering if I’m missing out on that in the way that you’re missing out on more vivid visual mental imagery.
How is classifying Platonic solids in 4 dimensions a reasonable test of mental imagery?
Summary: Thinking about geometry problems gives me access to visual mental imagery of lines, sometimes pretty stable and controllable lines. A few at a time.
Dodecahedron, stream of thought: 20 sides, right? What’s the… oh you asked edges. 20, clearly. Wait a dodecahedron is 3D. Okay I get the question now. Um, it has 20 sides… they each join or rather each edge joins a pair… aren’t they pentagons? So each side, face I guess, has 5 edges, that’s 20x5 is a hundred edges counted twice for 50 edges. --- During this time I Imagined a vague ball-like thing and Imagined the face of it nearest me and knew that it was a pentagon. Now I’m going to look up the answer before commenting, but I promise I’ll leave all of this even if I’m wrong. Oh it has 12 faces, oops. Then it’s 12x5/2=30 edges instead.
Medians: I Imagine an isoceles triangle and Mental Image the two equal sides of that triangle. I Imagine the vertical median and Mental Image a dark vertical bar across my visual field (this is with eyes closed btw). The bar quickly morphs into a jagged dark lightning bolt thing and back again and things turn amorphous. Um, I don’t know why the medians would be concurrent. Medians split the side, right? So… base times height, they split the area in two… base times height over two rather, whatevs. Okay. Do I know anything else about medians? No… let’s have two medians, clearly they intersect somewhere. I Imagine a scalene triangle with its base horizontal, as on a whiteboard in front of me, Mental Imaging the base as a thick bright line and the two other sides as deformations in the static. I mean it looks like the medians come kinda close. I want to draw things though, I’m losing track and it would take a lot of effort to prove this in my head.
Altitude: Imagining a pyramid with a dotted line from the tip to the middle of a triangular base. Mental Image is the lines of a triangle base with three vertical lines coming up like it’s a triangular prism, they won’t go together to make a point but hey 6 lines is a lot at once, neat. Side length s says triangle altitude is sqrt(s^2-s^2/4)=(sqrt(3)/2) s. I… am not sure where the triangle comes from that I can get the tetrahedron’s altitutde from, though. I want to draw. I Imagine the dotted line and try to make a triangle but I have to explicitly check “what is this line?” rather than seeing it. That’s like part of a triangle altitutde… oh hey the base triangle, I got a symmetric three interior lines both Imagined and Mental Imaged and there are some isoceles triangles there. A needed unknown x, x again, and s. And clearly it’s 30/30/120. So also it’s 30/60/90 with x, (sqrt(3)/2) s-x, and s/2. So x is the hypotenuse and is 2/sqrt(3) times s/2. x is s/sqrt(3). I’ve got (and I’m Imagining, and my Mental Image is like the Imagining except hella distorted but hey it’s there!) then a triangle with one edge the edge of a face s, one the altitude, and one s/sqrt(3). The altitude then is sqrt(s^2-s^2/3)=sqrt(2/3) * s. Checking… yep.
Platonic solids: Um. Four dimensions, huh? I Imagine a cube. Now it’s stretched and I Mental Image the static elongating, which lasts for not long. A hypercube must work, right? What is a four dimensional Platonic solid. It’s a 4D thing with regular 3D things as “faces”? Okay… how the hell does that work. If I can take a 3D thing, morph it over time until it’s back to a 3D thing, and interpret those morphs as… the same 3D thing? That doesn’t make sense, the morphs will be 1D. I am confused. I will look up four dimensional Platonic solids now. Okay, confusing.
This is great. More stream of consciousness while Guy solves math problems please.
I thought it was interesting that it was easier for me to picture the proper shapes than it was for you (I had no trouble getting the lines of my pyramid to join together, and I could easily imagine where the line for the altitude of the tetrahedron went), but you thought of the relations between line segment lengths and came up with the formulas for them much more quickly than I would have.
One thing I want to clarify though, when you said you were imagining the pyramid and dotted line, and then your mental imagine didn’t match that correctly—were you first successfully imagining the pyramid and dotted line, and then trying to also have a mental image, or when you said you were imagining did you just mean that you were starting to form the mental image? And if the former, what did this imagining consist of, other than just awareness of the abstract idea that an altitude should go from a face to its opposing point?