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?
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?