Concepts are learned. A concept to someone who has not learned it is meaningless.

So a person listening to the Mahler symphony from a culture that has not learned that musical language would probably not be so bothered by the intrusions of the rap song (which presumably he has not learned either). He would not know there were conceptual differences unless he started honing in on characteristics like the drums & bass pulsing in the rap and noticing that this was not present in the Mahler and going on from there.

In fact, on a pre-conceptual level, he might even think it was part of the show. He might even enjoy it.

By "thinking," I mean, in this context, specifically problem solving, which I see no way to do without using words.

Monkey in a cage. (The cage bars are widely enough spaced, the monkey can extend an arm through the openings.) Banana outside the cage, beyond the monkey's reach. Stick inside the cage.

Monkey uses stick to pull the banana to within its reach.

Would you call that "problem solving"?

For example, if I say: All men are mortal, then, as the second premise, I insert an image of Socrates, there is no way I can conclude that Socrates is mortal.

I don't see why not.

I'd appreciate it if someone would explain, preferably with examples, the meaning of "thinking in images." It's clear to me, of course, that during a process of problem-solving one may employ images as a means of concretizing, but -- if you do mean that one can problem-solve in images -- it's not at all clear to me that this can be done.

A simple example:

Are these figures the same shape with a different scale?

Apparently I'm using the term "thinking" to mean something quite different than those of you who say we can think in images. By "thinking," I mean, in this context, specifically problem solving, which I see no way to do without using words. For example, if I say: All men are mortal, then, as the second premise, I insert an image of Socrates, there is no way I can conclude that Socrates is mortal.

I'd appreciate it if someone would explain, preferably with examples, the meaning of "thinking in images." It's clear to me, of course, that during a process of problem-solving one may employ images as a means of concretizing, but -- if you do mean that one can problem-solve in images -- it's not at all clear to me that this can be done.

Barbara

Problem: What is the intersection of a cylinder with a plane not parallel to the axis of the cylinder?

Solution 1. Write down equations of plane and cylinder and solve. Result: Equation for ellipse.

Solution 2. Visualize the cylinder and plane intersecting. Result: An ellipse

Solution 2 is pure visual imagination. No words. Quicker, slicker.

In the visual solution, Bob, I get only an oval, not specifically an ellipse. With the equations (i.e., with natural language), we get exactitude.

On my shelf, a book not yet read is

After Euclid: Visual Reasoning and the Epistemology of Diagrams

Jesse Norman

CSLI 2006

From the back cover:

What is it to have visual intuition? Can we obtain geometrical knowledge by using visual reasoning? And if we can, is this because we have a faculty of intuition?

This book addresses these questions. It shows how mainstream philosophers since Leibniz have wrongly ignored visual reasoning as a source of knowledge; and how even basic geometrical reasoning that uses diagrams can be explained without using any appeal to a faculty of intuition. In so doing, this book helps to rehabilitate an ancient but long-disregarded tradition as it presents the first detailed philosophical case study of that branch of mathematical reasoning.

In Objectivity there are discussions of thought without language here:

Ba'al and Stephen are both partly correct. My visual solution follows. (I didn't do the math.) Using the three dimensional Cartesian coordinate system, imagine the center of the cylinder is the z-axis and the plane intersects the cylinder's perimeter entirely. If the plane is parallel to both the x-axis and y-axis, the intersection is a circle. If the plane is tilted relative to only the x-axis or only the y-axis, then the intersection is an ellipse. If it is tilted relative to both, then the intersection is an oval.

Ba'al and Stephen are both partly correct. My visual solution follows. (I didn't do the math.) Using the three dimensional Cartesian coordinate system, imagine the center of the cylinder is the z-axis and the plane intersects the cylinder's perimeter entirely. If the plane is parallel to both the x-axis and y-axis, the intersection is a circle. If the plane is tilted relative to only the x-axis or only the y-axis, then the intersection is an ellipse. If it is tilted relative to both, then the intersection is an oval.

If I'm understanding Merlin's description correctly, then I disagree with his conclusion. If the plane is tilted relative to both x and y, the intersection with the cylinder would be no different than when titling the plane relative only to x or only to y. The visual proof of this would be that, after tilting the plane relative to both x and y, one could rotate the Cartesian coordinate system on its z-axis around the cylinder to a point where the plane was angled relative only to x or to y.

Ba'al and Stephen are both partly correct. My visual solution follows. (I didn't do the math.) Using the three dimensional Cartesian coordinate system, imagine the center of the cylinder is the z-axis and the plane intersects the cylinder's perimeter entirely. If the plane is parallel to both the x-axis and y-axis, the intersection is a circle. If the plane is tilted relative to only the x-axis or only the y-axis, then the intersection is an ellipse. If it is tilted relative to both, then the intersection is an oval.

Merlin,

For some reason I cannot open the link. However, this post does very well illustrate the point of thinking in images, or more precisely, thinking in models built from elements abstracted from experience.

When I picture a cylinder intersected by a plane parallel only to the x-axis, I agree we have an ellipse. As one tilts the plane from the point where it is parallel with both the x and y-axis, one can picture a systematic deformation of the original circle that is equaled on both sides of the x-axis, thus creating the properties that distinguish the ellipse from a common oval. I do not see, however, how things change when one tilts the plane relative to both the x and y axes. This result can be created simply by turning the ellipse that is created by intersecting the cylinder by a plane parallel only to the x-axis around the z-axis. The same ellipse now has the z-axis at its centre and is intersect by a plane that is neither parallel to the x or y-axis. Non-elliptical ovals are not created by intersecting a cylinder with a plane.

Let me know if I am mistaken.

Either way, this process definitely demonstrates the idea of thinking in images but communicating in words.

Paul

Oops! It seems Jonathan beat me to it. Stop to put the kids to bed and loose my thunder.

[...] this process definitely demonstrates the idea of thinking in images but communicating in words.

Yes. Meanwhile...

I thought of an everyday sort of example to which Barbara might immediately relate: Choosing what clothes to wear. (Barbara is noted for being skilled in her choice of clothes.)

Barbara, although you might have a verbal accompaniment in the process of deciding what to wear, would you say it's really the verbalization which is the key factor in the thought process or instead your visual sense of what looks good with what? I.e., isn't the problem to be solved one which is solved much more by visualizing than by verbalizing?

Ba'al and Stephen are both partly correct. My visual solution follows. (I didn't do the math.) Using the three dimensional Cartesian coordinate system, imagine the center of the cylinder is the z-axis and the plane intersects the cylinder's perimeter entirely. If the plane is parallel to both the x-axis and y-axis, the intersection is a circle. If the plane is tilted relative to only the x-axis or only the y-axis, then the intersection is an ellipse. If it is tilted relative to both, then the intersection is an oval.

My words here are merely the "wrapping paper."

A right cylinder is circularly symmetric around its bore-line axis (the z axis). The choice of x or y axes are arbitrary

Ba'al and Stephen are both partly correct. My visual solution follows. (I didn't do the math.) Using the three dimensional Cartesian coordinate system, imagine the center of the cylinder is the z-axis and the plane intersects the cylinder's perimeter entirely. If the plane is parallel to both the x-axis and y-axis, the intersection is a circle. If the plane is tilted relative to only the x-axis or only the y-axis, then the intersection is an ellipse. If it is tilted relative to both, then the intersection is an oval.

My words here are merely the "wrapping paper."

A right cylinder is circularly symmetric around its bore-line axis (the z axis). The choice of x or y axes are arbitrary

Ba'al Chatzaf

You view this as intuitively obvious, without use of words?

Ba'al and Stephen are both partly correct. My visual solution follows. (I didn't do the math.) Using the three dimensional Cartesian coordinate system, imagine the center of the cylinder is the z-axis and the plane intersects the cylinder's perimeter entirely. If the plane is parallel to both the x-axis and y-axis, the intersection is a circle. If the plane is tilted relative to only the x-axis or only the y-axis, then the intersection is an ellipse. If it is tilted relative to both, then the intersection is an oval.

My words here are merely the "wrapping paper."

A right cylinder is circularly symmetric around its bore-line axis (the z axis). The choice of x or y axes are arbitrary

Ba'al Chatzaf

You view this as intuitively obvious, without use of words?

Bill P (who has a PhD in a mathematical field)

If I recall you are a specialist in statistics?

I don't always need words now. I also can visualize hyperbolic geometry in my head. It took several years to figure out how to do it. It is like riding a bike. One you catch on, you don't have to verbalize anymore.

I might have started out with clunky words and equations, but that is not how I ended up. My first action is to visualize and analogize. If that is insufficient then I do it the hard way, with mathematical symbolism. For proofs, I always use the verbal method for two reasons:

1. To communicate with others. Visualization is a private thing.

2. To make sure I have not overlooked something or assumed something not in the hypothesis.

On the other hand, some things cannot be visualized. Try visualizing a six dimensional Calabri-Yao manifold (this is used in String Theory). Good luck!

PS. Here is a visualization exercise for you, Cut a cube with a plane so that the intersection is a hexagon. Fun, fun, fun!

A right cylinder is circularly symmetric around its bore-line axis (the z axis). The choice of x or y axes are arbitrary

I already agreed to that (post 34).

Maybe Stephen thought of a slice through the cylinder that wasn't flat. By curving the slice some, the resulting surface could be an oval, but it would not be a flat oval nor an ellipse.

Here is a visualization exercise for you, Cut a cube with a plane so that the intersection is a hexagon. Fun, fun, fun!

That's easy. Click on the MPEG or QuickTime link here to view it.

Ba'al and Stephen are both partly correct. My visual solution follows. (I didn't do the math.) Using the three dimensional Cartesian coordinate system, imagine the center of the cylinder is the z-axis and the plane intersects the cylinder's perimeter entirely. If the plane is parallel to both the x-axis and y-axis, the intersection is a circle. If the plane is tilted relative to only the x-axis or only the y-axis, then the intersection is an ellipse. If it is tilted relative to both, then the intersection is an oval.

My words here are merely the "wrapping paper."

A right cylinder is circularly symmetric around its bore-line axis (the z axis). The choice of x or y axes are arbitrary

Ba'al Chatzaf

You view this as intuitively obvious, without use of words?

Bill P (who has a PhD in a mathematical field)

If I recall you are a specialist in statistics?

(snip)

Ba'al Chatzaf

With substantial background in Logic/Mathematical Foundations and Point Set Topology.

Some of our best intuition is kinesthetic. We have the ability to "think with our muscles".

Example 1. Riding a bike. The ability to ride a bike is gotten non verbally. One must get up on the bike and literally acquire a "feel" and learn to compensate to maintain balance. It is non verbal in toto. Try -telling- someone how to ride a bike. It cannot be done.

Example 2. Shagging a fly ball to the outfield. Human beings have the ability to predict ballistic trajectories over a limited range. Our ancestors had to learn to throw rocks (an later) spears to survive. The bility to visualize a trajectory is partly learned and partly inherited. As a result, a ten year old kid, with not a differential equation to his name, can experly shag fly balls, even running back to catch a long one. We have the ability to visualize a parabolic coruse.

Example 3. Kentucky windage. A good marksman learns how to compensate for wind and distance. He doesn't need mathematics. Mathematics comes in for long range naval gunnery where one must compensate for the Coriolis effect. But for short range just eye and practiced brain will do, with nary a word. During the American Revolution marksmen with Pennsylvania long rifles were picking off British officers at 200 yards. Most of them could barely read and none of them had any applicable mathematics.

Merlin, I was using the term oval to mean a plane closed figure that is oblong and curvilinear. When I visualize the intersection of a plane with a cylinder, where the plane is not perpendicular the axis of the cylinder, the intersection-figure I get is only so specific as oval in that sense of the term. I cannot see that it is an ellipse as opposed to some other oblong curvilinear figure.

In another case Bob mentioned, the case of visualizing a plane intersecting a cube such that the intersection is a hexagon, yes. There I get specifically a hexagon.

I mentioned to Larry about this thread being in process, and he said, as he has said a number of times before:

"A lot of times when I'm thinking physics, I hardly use words at all."

Ellen

___

I do something somewhat like "thinking in images", but I can never be sure how much if any of it makes sense until I can reduce it to words and/or mathematical notation. (I could add something about physicists not caring whether it makes sense, but that might be sarcastic, so I won't.) -- Mike Hardy

Ba'al and Stephen are both partly correct. My visual solution follows. (I didn't do the math.) Using the three dimensional Cartesian coordinate system, imagine the center of the cylinder is the z-axis and the plane intersects the cylinder's perimeter entirely. If the plane is parallel to both the x-axis and y-axis, the intersection is a circle. If the plane is tilted relative to only the x-axis or only the y-axis, then the intersection is an ellipse. If it is tilted relative to both, then the intersection is an oval.

My words here are merely the "wrapping paper."

Merlin, if it's tilted relative to both, then it's an ellipse. I don't know what you mean by an "oval", but the shape of the intersection depends only on the norm of the gradient of the plane, and not on the direction of the gradient. -- Mike Hardy

Merlin, I was using the term oval to mean a plane closed figure that is oblong and curvilinear. When I visualize the intersection of a plane with a cylinder, where the plane is not perpendicular the axis of the cylinder, the intersection-figure I get is only so specific as oval in that sense of the term. I cannot see that it is an ellipse as opposed to some other oblong curvilinear figure.

In another case Bob mentioned, the case of visualizing a plane intersecting a cube such that the intersection is a hexagon, yes. There I get specifically a hexagon.

The proof that it is an ellipse, in the sense of having two foci, such that the sum of the distances from a point on the curve to the two foci remains constant as the point moves along the curve, can be proved by a variety of methods, including that of Dandelin spheres. Dandelin spheres can also be used for the intersection of a plane with a cone. That's why the term "conic section" is used. Look up "Dandelin spheres" on Wikipedia. -- Mike Hardy

Some of our best intuition is kinesthetic. We have the ability to "think with our muscles".

We do that chronically in our waking motions. Probably the vast majority of our knowledge of physically navigating our environment is never put into words. Even with skills which we use verbal instructions to help us acquire, we have to get to the stage of the "knowledge" being in our muscles for the performance to be proficient -- and by that stage, if you try to state what you're doing while you're doing it, you'll likely mess up (like the proverbial centipede that couldn't walk when it tried to think about the order in which it moved its legs).

PS. Here is a visualization exercise for you, Cut a cube with a plane so that the intersection is a hexagon. Fun, fun, fun!

I immediately saw that, though I don't practice this stuff near as much as some others contributing to the thread, nor do I have near the knowledge of the formal geometry.

I'm pretty good at visualizing hyperbolic geometry, too, though I couldn't begin to say what most of the equations are.

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## Michael Stuart Kelly

Here is an addition to my last post.

Concepts are learned. A concept to someone who has not learned it is meaningless.

So a person listening to the Mahler symphony from a culture that has not learned that musical language would probably not be so bothered by the intrusions of the rap song (which presumably he has not learned either). He would not know there were conceptual differences unless he started honing in on characteristics like the drums & bass pulsing in the rap and noticing that this was not present in the Mahler and going on from there.

In fact, on a pre-conceptual level, he might even think it was part of the show.

He might even enjoy it.(shudder...)Michael

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## Ellen Stuttle

Monkey in a cage. (The cage bars are widely enough spaced, the monkey can extend an arm through the openings.) Banana outside the cage, beyond the monkey's reach. Stick inside the cage.

Monkey uses stick to pull the banana to within its reach.

Would you call that "problem solving"?

I don't see why not.

A simple example:

Are these figures the same shape with a different scale?

AAEllen

___

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## BaalChatzaf

Problem: What is the intersection of a cylinder with a plane not parallel to the axis of the cylinder?

Solution 1. Write down equations of plane and cylinder and solve. Result: Equation for ellipse.

Solution 2. Visualize the cylinder and plane intersecting. Result: An ellipse

Solution 2 is pure visual imagination. No words. Quicker, slicker.

Ba'al Chatzaf

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## Guyau

In the visual solution, Bob, I get only an oval, not specifically an ellipse. With the equations (i.e., with natural language), we get exactitude.

On my shelf, a book not yet read is

After Euclid: Visual Reasoning and the Epistemology of DiagramsJesse Norman

CSLI 2006

From the back cover:

In

Objectivitythere are discussions of thought without language here:V1N114,16–19,V1N238–40, 48, 54–55,58–59,74–75,V1N36–9,33–34, 70–72,74–76, 81,V2N1113, 118,V2N4115, 228–29, 231One reference used by the

Objectivitywriters is the collectionThought without LanguageL. Weiskrantz, editor

Oxford 1988

A recent addition on this topic:

Thinking without WordsJose Luis Bermudez

Oxford 2008

http://www.oup.com/uk/catalogue/?ci=9780195341607

Edited by Stephen Boydstun## Link to comment

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## merjet

Ba'al and Stephen are both partly correct. My

visualsolution follows. (I didn't do the math.) Using the three dimensional Cartesian coordinate system, imagine the center of the cylinder is the z-axis and the plane intersects the cylinder's perimeterentirely. If the plane is parallel to both the x-axis and y-axis, the intersection is a circle. If the plane is tilted relative to only the x-axis or only the y-axis, then the intersection is an ellipse. If it is tilted relative to both, then the intersection is an oval.My words here are merely the "wrapping paper."

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## Jonathan

If I'm understanding Merlin's description correctly, then I disagree with his conclusion. If the plane is tilted relative to both x and y, the intersection with the cylinder would be no different than when titling the plane relative only to x or only to y. The visual proof of this would be that, after tilting the plane relative to both x and y, one could rotate the Cartesian coordinate system on its z-axis around the cylinder to a point where the plane was angled relative only to x or to y.

J

Edited by Jonathan## Link to comment

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## Paul Mawdsley

AuthorMerlin,

For some reason I cannot open the link. However, this post does very well illustrate the point of thinking in images, or more precisely, thinking in models built from elements abstracted from experience.

When I picture a cylinder intersected by a plane parallel only to the x-axis, I agree we have an ellipse. As one tilts the plane from the point where it is parallel with both the x and y-axis, one can picture a systematic deformation of the original circle that is equaled on both sides of the x-axis, thus creating the properties that distinguish the ellipse from a common oval. I do not see, however, how things change when one tilts the plane relative to both the x and y axes. This result can be created simply by turning the ellipse that is created by intersecting the cylinder by a plane parallel only to the x-axis around the z-axis. The same ellipse now has the z-axis at its centre and is intersect by a plane that is neither parallel to the x or y-axis. Non-elliptical ovals are not created by intersecting a cylinder with a plane.

Let me know if I am mistaken.

Either way, this process definitely demonstrates the idea of thinking in images but communicating in words.

Paul

Oops! It seems Jonathan beat me to it. Stop to put the kids to bed and loose my thunder.

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## merjet

I fixed the link in my prior post. http://en.wikipedia.org/wiki/Cartesian_Coordinate_System

Paul and Jonathan, I stand corrected, I think. I didn't think about it very long and didn't question Stephen.

So, Stephen, why did you say "I get only an oval, not specifically an ellipse"?

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## Ellen Stuttle

Yes. Meanwhile...

I thought of an everyday sort of example to which Barbara might immediately relate: Choosing what clothes to wear. (Barbara is noted for being skilled in her choice of clothes.)

Barbara, although you might have a verbal accompaniment in the process of deciding what to wear, would you say it's really the verbalization which is the key factor in the thought process or instead your visual sense of what looks good with what? I.e., isn't the problem to be solved one which is solved much more by visualizing than by verbalizing?

Ellen

___

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## Ellen Stuttle

I mentioned to Larry about this thread being in process, and he said, as he has said a number of times before:

"A lot of times when I'm thinking physics, I hardly use words at all."

Ellen

___

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## BaalChatzaf

A right cylinder is circularly symmetric around its bore-line axis (the z axis). The choice of x or y axes are arbitrary

Ba'al Chatzaf

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## Alfonso Jones

You view this as intuitively obvious, without use of words?

Bill P (who has a PhD in a mathematical field)

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## BaalChatzaf

If I recall you are a specialist in statistics?

I don't always need words now. I also can visualize hyperbolic geometry in my head. It took several years to figure out how to do it. It is like riding a bike. One you catch on, you don't have to verbalize anymore.

I might have started out with clunky words and equations, but that is not how I ended up. My first action is to visualize and analogize. If that is insufficient then I do it the hard way, with mathematical symbolism. For proofs, I always use the verbal method for two reasons:

1. To communicate with others. Visualization is a private thing.

2. To make sure I have not overlooked something or assumed something not in the hypothesis.

On the other hand, some things cannot be visualized. Try visualizing a six dimensional Calabri-Yao manifold (this is used in String Theory). Good luck!

PS. Here is a visualization exercise for you, Cut a cube with a plane so that the intersection is a hexagon. Fun, fun, fun!

Ba'al Chatzaf

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## merjet

I already agreed to that (post 34).

Maybe Stephen thought of a slice through the cylinder that wasn't flat. By curving the slice some, the resulting surface could be an oval, but it would not be a flat oval nor an ellipse.

That's easy. Click on the MPEG or QuickTime link here to view it.

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## Alfonso Jones

With substantial background in Logic/Mathematical Foundations and Point Set Topology.

Bill P (Alfonso)

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## BaalChatzaf

A topologist! Then you should have no problem visualizing a Klein Bottle in four dimensions.

Ba'al Chatzaf

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## Alfonso Jones

Point Set Topology a la R. L. Moore, in case the name means something to you.

But what a trained mathematician has learned to visualize may not be indicative of what is typically simply visual for others.

Bill P (Alfonso)

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## BaalChatzaf

Some of our best intuition is kinesthetic. We have the ability to "think with our muscles".

Example 1. Riding a bike. The ability to ride a bike is gotten non verbally. One must get up on the bike and literally acquire a "feel" and learn to compensate to maintain balance. It is non verbal in toto. Try -telling- someone how to ride a bike. It cannot be done.

Example 2. Shagging a fly ball to the outfield. Human beings have the ability to predict ballistic trajectories over a limited range. Our ancestors had to learn to throw rocks (an later) spears to survive. The bility to visualize a trajectory is partly learned and partly inherited. As a result, a ten year old kid, with not a differential equation to his name, can experly shag fly balls, even running back to catch a long one. We have the ability to visualize a parabolic coruse.

Example 3. Kentucky windage. A good marksman learns how to compensate for wind and distance. He doesn't need mathematics. Mathematics comes in for long range naval gunnery where one must compensate for the Coriolis effect. But for short range just eye and practiced brain will do, with nary a word. During the American Revolution marksmen with Pennsylvania long rifles were picking off British officers at 200 yards. Most of them could barely read and none of them had any applicable mathematics.

Ba'al Chatzaf

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## Guyau

Merlin, I was using the term

ovalto mean a plane closed figure that is oblong and curvilinear. When I visualize the intersection of a plane with a cylinder, where the plane is not perpendicular the axis of the cylinder, the intersection-figure I get is only so specific asovalin that sense of the term. I cannot see that it is an ellipse as opposed to some other oblong curvilinear figure.In another case Bob mentioned, the case of visualizing a plane intersecting a cube such that the intersection is a hexagon, yes. There I get specifically a hexagon.

~~~~~~~~

Conversational Context: #25, #29, #30, #34

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## Mike Hardy

I do something somewhat like "thinking in images", but I can never be sure how much if any of it makes sense until I can reduce it to words and/or mathematical notation. (I could add something about physicists not caring whether it makes sense, but that might be sarcastic, so I won't.) -- Mike Hardy

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## Mike Hardy

Merlin, if it's tilted relative to both, then it's an ellipse. I don't know what you mean by an "oval", but the shape of the intersection depends only on the norm of the gradient of the plane, and not on the direction of the gradient. -- Mike Hardy

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## Mike Hardy

The proof that it is an ellipse, in the sense of having two foci, such that the sum of the distances from a point on the curve to the two foci remains constant as the point moves along the curve, can be proved by a variety of methods, including that of Dandelin spheres. Dandelin spheres can also be used for the intersection of a plane with a cone. That's why the term "conic section" is used. Look up "Dandelin spheres" on Wikipedia. -- Mike Hardy

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## Ellen Stuttle

We do that chronically in our waking motions. Probably the vast majority of our knowledge of physically navigating our environment is never put into words. Even with skills which we use verbal instructions to help us acquire, we have to get to the stage of the "knowledge" being in our muscles for the performance to be proficient -- and by that stage, if you try to state what you're doing while you're doing it, you'll likely mess up (like the proverbial centipede that couldn't walk when it tried to think about the order in which it moved its legs).

Ellen

___

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## Ellen Stuttle

I immediately saw that, though I don't practice this stuff near as much as some others contributing to the thread, nor do I have near the knowledge of the formal geometry.

I'm pretty good at visualizing hyperbolic geometry, too, though I couldn't begin to say what most of the equations are.

Ellen

___

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