I’m just posting this on this cold morning because, when I visited the nearby coffee-and-bagel shop (Einstein’s) in the student union, I found a stack of those cardboard cup-holders designed to prevent you from burning your hands and suing the shop. I took two of them for the photos below. I seem to remember that Matthew put up this illusion recently, but now anyone can do it who has access to a fancy-schmancy coffee shop.
These two holders are the same size, but they don’t look it, do they?
Don’t believe it? Here they are stacked atop each other:
With this trick you can be the life of the party—assuming the party is held in a coffee shop.
This is an illusion, but if you think geometrically, and experiment with extreme cases, it is easy to ‘see’:
As to the “experiment(s)”, what these each are amount to a medium angle’s part of a moderate annulus—annulus being that portion of a disc between a smaller and a larger radius. The illusion completely disappears in any of several extreme cases—make the two radii very nearly equal, or do the opposite and make the smaller radius nearly zero; or finally, make the angle much closer to the full 360 deg, rather than the moderate, approximately 30 deg.
The reason for the illusion is that our subconscious says what you have here are two things which are so close to being rectangles that we may as well take them to be that, i.e. not to have any curvature at all.
Sorcery!
Miracle of God!
That is so cool.
Watch out, now the coffee shops will start charging more for the bottom one.
There’s a bit of a “cheat” going on here (unintentional, I’m sure) that significantly enhances this illusion.
Looking at the picutre, my eye suggests the seemingly smaller one is somewhere between 5%-10% smaller.
BUT, if you actually measure the image, it *is* about 4-5% smaller. Do it. Get a ruler and measure corner to corner. The “smaller” one actually is smaller, not just an illusion.
This is the result of the camera angle, with the lower one being slightly closer to the camera. Because the image is 2D, losing depth perception you would have in life, about half of the perceived size difference is a *real* size difference rather than an optical illusion.
If you viewed the same arrangement “live”, with two eyes, your depth perception would compensate for the difference in depth angle, making the optical illusion less dramatic.
Thus the “illusion” itself only accounts of a bit less than a 5% effect, not the 10% effect we seem to get here.
Yes. I also made the measurements, because they didn’t look the same at a quick view.
But what about the lens effects? Digital cameras have variable coverage (wide angle vs. narrower coverage). Could that be more important here?
Perhaps someone with expert knowledge on lenses could weigh in.
If by lens effect you mean perspective and relative image size due to differing depths, then that is quite likely what is going on here.
Basically, for small (percent) variations in distance, the image size will be inversely proportional. So if the lower cup holder is 3% closer to the lens, then it will appear 3% bigger (note for small changes, such as 3%, 1/1.03 ~= .97).
The optical illusion is still valid, but I think that extra ~3% or so is kicking it up a notch.
I believe you can easily make versions of this, and where you are look straight down at the middle of a very homogeneous, no printing, example, as well as photographing it that way. And then do the same with the earlier ‘smaller’ one underneath.(Here make the two have different solid colours, so you can interchange their positions ‘meaningfully’.) Now turn it (or you or the photo) around, so to speak, so it is concave upwards rather than downwards.
In every case, the optical illusion is still there, very dramatically.
So I largely disagree with this, to the extent it is even meaningful to give percentages.
Here’s a modification I did with Paint. I made room at the top of the image, then copied and pasted the bottom cup holder there. The top and bottom cup holders are exactly the same image, pixel-for-pixel:
http://imgur.com/I1pa2By
I measured lower left corner to upper right corner on my screen for all three images: Top = 9.55 cm. Middle = 9.25 cm. Bottom = 9.55 cm.
I measured top left corner across to top right corner on my screen for all three images: Top = 9.1 cm. Middle = 8.9 cm. Bottom = 9.1 cm.
Conclusion: middle image is smaller.
Perhaps this is due one or both of the following:
(1)the lens not being parallel to the plane of the objects when the photo was originally taken
(2)the center of the lens not being centered between the objects when the photo was originally taken
It is perhaps worth pointing out that photocopiers have problems producing images that are fully proportional and accurate renditions of the originals; straight lines along the edges of the copying fields are often slightly curved, for example.
Excellent!
To my eye (though now one could reasonably say a bias plays a role in my perception) the new “top” cup holder looks about the same size as the former top (now middle) cup holder.
Since we know it is actually larger, the optical illusion is having its effect.
I do think the additional (maybe 3%? given the measurements by Karst) adds significantly to how dramatic it appears to be.
Nice, thanks!
Always good to be reminded how fallible perception is.
Are you sure that wasn’t Jastrow’s bagel shop you visited?
Lay them out like in the photo and ask “which one’s bigger?”
When you’re told it’s the bottom one, pick up the top one and act like you’re pulling and stretching it, then lay it down under the other one.
Then challenge the other person to stretch the top one to match.
I don’t know if I’d call this a true “optical” illusion- it’s more of a “brain-assumption” illusion- the placement is what causes it: if both cup holders were centered, one above the other, they would appear the same size. “Scooting” the lower one to the right so its upper corner lines up with the LOWER corner of the top one “doubles” the width difference between the top and bottom of the cup holder on the right side. Two lines drawn along the left and right edges (the right line connecting the top corners of the cup, or even the bottom corners) would be parallel, indicating that the width of the holders is the same.