Math Is Fun Forum

paulb203

Member

Registered: 2023-02-24

Posts: 136

How do I know if I've successfully drawn a cube?

If I draw a square, 2cm by 2cm, then draw another square, also 2cm by 2cm, starting the second from the centre of the first, in the manner children attempt to draw their first cube (and in the manner I still occasionally try to draw a cube and wonder if it’s actually a cube. Then I join the corners of the two squares in the same manner, have I actually drawn a cube?

When I look at the finished drawing it appears to be a cube. But the only measurements I’m certain of are the respective lengths and breadths of the two squares, each 2 by 2.

The four lines used to join the two squares, to give the cube its depth, are less than 2cm, on the page. If it was an actual cube in front of me, an actual 3d solid shape, those ‘depth’ lines would measure 2cm also, obviously. Does it matter that the depth lines are not 2 cm on the page? Is it still a cube?

phrontister

Real Member

From: The Land of Tomorrow

Registered: 2009-07-12

Posts: 4,823

Re: How do I know if I've successfully drawn a cube?

...have I actually drawn a cube?

When I look at the finished drawing it appears to be a cube.

See post #9 for why (via Pythagoras) BF ≈ 1.414cm long instead of the 2cm that it should be.

Looks to me like it's a perspective conundrum, as with some optical illusions.

In the image below (which I drew in Geogebra), the left-hand image looks like a cube (but the measurements tell me it's not), and the right-hand image is a cube according to the measurements (but it certainly doesn't look like one)!

I even printed the image to check the measurements, got my ruler out, and..........they're all correct!

There are other illusions, too: eg,
   1. The green and red squares are congruent, but the green square seems to be larger than the red square.
   2. Lines AF & BG are parallel, but don't seem to be.
   Similarly with the right-hand image...

Re the congruent-squares illusion: I guess that my brain's probably thinking perspectively, and 'knows' that in these 3d images the rear square should appear smaller than the front square. With that not being the case (the squares being congruent), my brain then 'sees' the rear square as larger than the front square. Similarly with the 'non-parallel' lines...
 

Last edited by phrontister (2023-10-11 12:49:04)

---

"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

Bob

Administrator

Registered: 2010-06-20

Posts: 10,196

Re: How do I know if I've successfully drawn a cube?

The human brain 'learns' about the world by taking in information using the senses.  Visually, we learn what a cube is by seeing cubes (and non-cubes) and looking at them from all angles.  As the back end of a cube makes a smaller image on the retina our brains 'learn' that that is what a cube looks like.  When we try to represent that 3D object in 2D the brain recognises that the equal square version is 'wrong' because the back square should look smaller.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

phrontister

Real Member

From: The Land of Tomorrow

Registered: 2009-07-12

Posts: 4,823

Re: How do I know if I've successfully drawn a cube?

You make good sense, Bob!

---

"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

phrontister

Real Member

From: The Land of Tomorrow

Registered: 2009-07-12

Posts: 4,823

Re: How do I know if I've successfully drawn a cube?

Amazingly, if you convince your brain that the back wall is the front wall and the font wall is the back wall, the new back wall becomes the larger of the two!!! For both images.

ie, the wall that was previously the smaller, becomes the larger!!! What!?

Some optical illusions behave the same way.

Just to clarify, I'd drawn the images so that the red and the blue walls are the front walls to the left of, and lower than, the respective back walls. This new illusion has the green and the brown walls as the front walls to the right of, and higher than, the respective back walls.

Cool!

Last edited by phrontister (2023-10-08 00:06:35)

---

"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

Bob

Administrator

Registered: 2010-06-20

Posts: 10,196

Re: How do I know if I've successfully drawn a cube?

Nah, I'm not getting that. On the right it now seems like a shape that is getting larger towards the back.  I wondered if this means our brains are different so I turned my laptop upside down to see if an Oz perspective changed anything.  It didn't!

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

phrontister

Real Member

From: The Land of Tomorrow

Registered: 2009-07-12

Posts: 4,823

Re: How do I know if I've successfully drawn a cube?

I wondered if this means our brains are different so I turned my laptop upside down to see if an Oz perspective changed anything. It didn't!
Bob

Nope, that wouldn't work; both you and the laptop have to be upside down (ie, from your perspective up where you are).   That's how we do it down here.

That way (the correct way), if you drop your laptop, it falls up to the ground (where you could find it again...albeit in bits & pieces) instead of down to the sky (from where it would be irretrievable). That's from your inverted perspective, of course.

Nah, I'm not getting that. On the right it now seems like a shape that is getting larger towards the back.Bob

A couple of things you could try (for simplicity with referencing, I'll refer to the left-hand image, but the same would apply to the other one):
1. (a) Focus on the little square (ie, the region where the front and back walls overlap).
    (b) Concentrate on that and let your brain relax into neutral (to make it believe you're not up to any tricks).
    (c) Casually – and maybe even humming or whistling nonchalantly (the brain musn't suspect anything) – observe peripherally the role reversal of the green and red squares: ie, that the green is now the front, higher, smaller square, and the red is now the rear, lower, larger square.
    (d) Squinting may help (vary squinting severity if you're getting nowhere with it).

2. Using paulb203's method from the first paragraph of post #1, construct the left-hand image to look like mine.
    However, whereas I started with the red image as the front wall, followed by the green image as the back wall to the right, and higher than, the front wall, try starting with the green image as the front wall, followed by the red image as the back wall to the left, and lower than, the front wall.

Enjoy!

Last edited by phrontister (2023-10-08 22:30:04)

---

"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

Bob

Administrator

Registered: 2010-06-20

Posts: 10,196

Re: How do I know if I've successfully drawn a cube?

I'm not having difficulty changing which face is in front.  For the left hand image the two faces look ok but the connecting lines seem too long.  The connecting lines also seem too long for the right hand image. Because the blue square is bigger my brain says this isn't a cube at all but rather the frustrum of a square based pyramid with me looking at the top, cropped off, face, and then getting bigger towards the base at the back.

I've got a better idea.  Let's design some 3D paper so we can draw this properly without all these 2D limitations. Oh yes, we can do that by making a cube out of card.  Back face looks smaller.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

phrontister

Real Member

From: The Land of Tomorrow

Registered: 2009-07-12

Posts: 4,823

Re: How do I know if I've successfully drawn a cube?

If I draw a square, 2cm by 2cm, then draw another square, also 2cm by 2cm, starting the second from the centre of the first, in the manner children attempt to draw their first cube (and in the manner I still occasionally try to draw a cube and wonder if it’s actually a cube. Then I join the corners of the two squares in the same manner...

Sorry, paulb203, I should have posted this earlier:

So what I've done there is to find BF by using Pythagoras, after working out the lengths of KF and KB.

...have I actually drawn a cube?

No, because the lines that join the corners of the two squares are 1.414cm long instead of the 2cm that they should be (see the above Pythagorean proof).

My 2d representation of a 3d cube may look like a cube, but it actually isn't one: it's just that the connecting lines lead our minds to believe that it is (see posts 2 & 3).

Last edited by phrontister (2023-10-10 23:38:51)

---

"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

paulb203

Member

Registered: 2023-02-24

Posts: 136

Re: How do I know if I've successfully drawn a cube?

Thanks, everybody.

Another one that was way more complicated than I thought it would be

Got to love maths!