Why does the brain make such erroneous assumptions?

Flavus Aquila said:
There are triangles within a circle, however, a circle is not defined by triangles. Concretely I would say that there are potentially triangles in a circle, but not formally.

The notion/form of a triangle within a circle is an abstract - litterally abstracted by the intellect, from material relations. This abstract can then be imposed on matter, by for example, drawing a triangle within a circle. However, this drawing will only be an imperfect representation of a perfect, abstracted intellectual form of triangle.

I don't like the term "ideal form" because of it's platonic associations, as though there were really such a thing as triangle-ness outside the intellectual notion of what a triangle is.

The intellectual notion of a triangle is universal, insofar as it comprehends all three sided, three angled closed figures/shapes/spatial-relationships. It is not ideal, so much as abstracted from all particular material examples.
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Well ideal triangles don't necessarily exist - It might not be impossible for a perfect triangle to actually exist, rather what is probably true is that perfect triangles are simply unlikely.

However, ideal triangles are useful. Greatly useful in structural engineering for example, since a triangle is one of the strongest shapes you can have. So generally the closer you can get it to ideal, the better off you are, but things are almost always balanced mostly by budget, materials, time, effort, and the realm of possibility in general.
 
Also note that I'm pragmatic.

If something is only worthwhile in theory, and can't be condensed down to practice, I just kind of shrug about it. Unless it's for entertainment value.
 
only read the first post;
perhaps it has to do with top-down knowledge.

perhaps that even when we know the rules for a three dimensional perspective, our brain might not understand what a cube is, and thus just connecting the lines, seing it as a three dimensional thing, rather than a cube (it could be 3d, while not being a perfect cube).

have you seen the hollow mask illusion?
 
Elis said:
only read the first post;
perhaps it has to do with top-down knowledge.

perhaps that even when we know the rules for a three dimensional perspective, our brain might not understand what a cube is, and thus just connecting the lines, seing it as a three dimensional thing, rather than a cube (it could be 3d, while not being a perfect cube).

have you seen the hollow mask illusion?
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[video=youtube;XyALAuKiKug]http://www.youtube.com/watch?v=XyALAuKiKug[/video]

I have seen it. However, false assumptions are more justified in this case since the object is actually designed to trick you. It has a physical basis which operates differently.

However, one issue with the Necker Cube is that it has a reputation for being a cube. This is confounded by the fact that most people inherently see perspective, but must learn to duplicate it in their drawings. This is evidenced by the fact that children's drawings are incredibly simplified and abstract.

Somebody I talked to about art once said "Draw what you see, not what you think you see." This is difficult for many to distinguish because one emulates reality, while the other only represents it.
 
I wonder how a child would interpret the "cube".

Someone who have never been presented cubes, or any other symetric shapes.
 
Elis said:
I wonder how a child would interpret the "cube".

Someone who have never been presented cubes, or any other symetric shapes.
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Good question. There should be a study on this.

I'm trying to look for data, but haven't found much on how children perceive the Necker Cube, other than one study showing that children report less perspective flipping than adults do. This is kind of odd as I found results for a study done with the Necker Cube on monkeys which shows that monkeys do not perceive the illusion. I think it's a little absurd that I can find this study more easily than children's studies.

However, Wikipedia has one anecdotal case, Sidney Bradford, who was blind at 10 months but gained vision through surgery after 52 years. Bradford did not fall for the Necker Cube illusion.
 
A perfect triangle is simple. It is a form of a triangle. Everything is equal, as in a square.

Speak for your own brains.
 
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just me said:
A perfect triangle is simple. It is a form of a triangle. Everything is equal, as in a square. Speak for your own brains.
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Equality is not what makes a triangle be a triangle though - ratio is. Hence, the angles of a triangle add up to 180 degrees, and because of this fact, trisectoring any triangle makes a 'perfect' equilateral triangle.

It's called Morley's trisector theorem.
 
[MENTION=680]just me[/MENTION]

Also why should equality make something perfect? Sounds arbitrary to me. What makes an equilateral triangle be a perfect triangle in your opinion, and why is it useful? What's special about it?

What I call an ideal triangle is simply a triangle that has sides which are not warped in any way, so that it follows the rule that all the corners add up to 180 degrees precisely. This is an accurate description of ideal precision, and it has something to be measured against. If you have a triangle with bent sides, the angles will add up to more or less than 180 which would induce error in calculations.
 
sprinkles said:
@just me

Also why should equality make something perfect? Sounds arbitrary to me. What makes an equilateral triangle be a perfect triangle in your opinion, and why is it useful? What's special about it?

What I call an ideal triangle is simply a triangle that has sides which are not warped in any way, so that it follows the rule that all the corners add up to 180 degrees precisely. This is an accurate description of ideal precision, and it has something to be measured against. If you have a triangle with bent sides, the angles will add up to more or less than 180 which would induce error in calculations.
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Why a perfect cube, or square, or even a circle? Why would inequality make something perfect?
 
just me said:
Why a perfect cube, or square, or even a circle? Why would inequality make something perfect?
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In Euclidean geometry, Euclid based all angles on the square, or the right angle. Other angles are fractions or ratios of this angle, and this is where the mathematical concept of "square" comes from - in Pythagorean theorem for example, they literally used squares to prove it, but now we can represent a square by saying "something times itself", which actually represents the area of a square.

So, squares are not squares because they are equal, but rather they are squared because they are square, and have 90 degree corners.

Edit:
Also the concept of cube was similarly derived from this. Same principle with an added dimension, a value to the third power. n^3 came from the idea of a cube, not vice versa.
 
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Additionally, a perfect circle results from invariance. If the radius or diameter is invariant in a given circle, then it is necessarily perfectly circular, and these values will necessarily be equal in all directions.

So in some cases equality arises from invariance, but invariance does not always imply some form of equality.
 
Furthermore, concerning invariants, you can infer things from them.

For example, if you must store whole 10 pies among only 4 rooms, and each room must contain at least the entirety of 1 whole pie, and can contain no other pies other than a pie from these 10, you can infer some things such as:

A given room can have no more than 7 pies.
At least one room must have more than 2 pies.
If somebody eats a pie, you no longer have 10 whole pies - hope the universe doesn't implode.
 
just me said:
Is it perfect or ideal; just another form?
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Rectangle-ness is an ideal form. Which is why you go through so much trouble to make one.

This is also evidenced by the fact that classical geometry construction uses an unmarked straight edge and a compass. Using any other tools was considered improper and not rigorous. In modern times we can accept that using a ruler and marking out 10cm or whatever and squaring it up with a protractor is acceptable, but this is also less rigorous and more prone to error.

So a perfect rectangle is perfectly rectangular, and a perfect square is perfectly square. A perfect square is also a perfect rectangle because squares are also rectangles.

Edit:
Or in other words all lines were considered infinite, and measuring them with marks was not allowed. So any equality did not result from comparing sides and ensuring they were equal. i.e. squareness was not determined by equality of the sides since you're not allowed to measure length. Equality must result from the precision of the construction, and not be a factor in the construction process.
 
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