Math Professor Invents Non-Reversing Mirror | Hacker News Reader

Math Professor Invents Non-Reversing Mirror

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154 pointsColinWright78 comments

> When you look in a normal mirror, the image you see of yourself is in reverse.

Funny thing: Mirrors don't reverse left and right. They reverse front and back.

So this "non-reversing" mirror is actually mapping y to -y, x to -x (Where the mirror is in front of you in the z-direction.

Curious.

Edit: I believe tomerv below is correct: x is mapping to -x but y is kept the same.

If you map both x to -x and y to -y (in addition to mapping z to -z), then you still get a reversed (chiral) image. From the pictures, it looks like it maps x to -x, but y is kept the same.

Demonstration time:

Write something on a piece of paper. Hold the paper up to the mirror.

Now replace that with a transparent piece of plastic.

What is that supposed to demonstrate?

That people on HN have too much spare time.

Ob. xkcd http://xkcd.com/1145/

See ALT text.

https://www.youtube.com/watch?v=wrhYOYs5JrA

I still don't intuitively understand how reversing front to back results in what we perceive as a left/right inversion.

It took me a long time to figure this out as a child, since nobody I asked knew the answer! Is it something about the vertical of the mirror being different than the horizontal? No, if you rotate the mirror 90 degrees it still inverts left/right and not top/bottom. Is it something about the Earth that makes horizontal different from vertical? No, if you look into a mirror while your body is in horizontal position, it still inverts left/right, and not top/bottom (viewed from your body). Is it something about our two eyes being on a horizontal line instead of a vertical line? No, if you close one eye it still works all the same. If you trace out how the light goes, you'll see that there is nothing different about horizontal vs vertical. Yet clearly a mirror inverts things horizontally but not vertically. How can this be?

The real answer is that we perceive this because we are used to turning in the horizontal plane. If you see yourself in the mirror and you raise your right hand, the guy in the mirror is raising his left hand. But how do you know that it is his left hand? Mentally you place yourself in his position by rotating yourself and then you check which hand it is, but by doing this rotation your left hand ends up in the position of your right hand in the mirror. So even though the alignment of you and your mirror image appears to be perfect on first sight, it actually isn't. If we were creatures that are used to only turning in the vertical plane, then we'd perceive the same image as top/bottom inversion. You'd mentally rotate yourself vertically, and then instead of your left hand ending up where you see your right hand and vice versa, your head would end up where you see your feet and vice versa.

The fact that our bodies are roughly symmetrical in one direction but not in the other makes this a bit non obvious unfortunately. It may be easier to understand this with an object that's not symmetrical in any direction. If you have a piece of paper with some word written on it, and you look at it via a mirror, why is the word flipped from left to right and not from top to bottom? This question has the same answer: because we're used to turning things in the horizontal plane. To show the paper to yourself via the mirror, you rotated the paper so that the text is now facing away from you, towards the mirror. But how did you rotate it? You rotated it by turning it in the horizontal plane. If you had rotated it in the vertical plane, the word would be flipped vertically. It's just that if you do this nobody is surprised that the word ends up flipped vertically, because that's not how our psychology works.

The real headscratcher comes when you realise that all reversions are equivalent disregarding rotation. That is, reversing left/right is the same as reversing up/down and rotating around z.

You can actually get a up/down mirroring effect (intuitively speaking) by standing on a mirror and holding a piece of paper perpendicular to it. It will look up/down reversed, not left/right.

It's still left right reversed though, just try it with something printed on the paper.

Since left and right and up and down are both the same, it's essentially as though you're looking through the back of the object. Imagine if you made a sign clear and looked through the back. Even though no reversal has taken place, the text looks backward. This is why things appear reversed.

If you are confused about how left and right are not actually switched, I would say the easiest way is to think in absolute terms, not relative. That is, don't think about left and right, think about east and west. Face north and paint your east (right) hand red. Now when you look at a mirror (you're still facing north), your east hand is red, but the reflection's east hand (think in real world terms here) is also red. That's because the reflection's horizontal aspects have not been reversed.

Richard Feynman enjoys discussing this: http://www.youtube.com/watch?v=msN87y-iEx0&gl=CA

Print out a word in a very large font on a piece of paper. Now, hold the blank side up to a light and try to read the word - it will be backwards. This is because you're reading the paper back to front.

Our mind makes it a left to right inversion because we first try to think of it as a rotation. Look in the mirror while holding a hat in your left hand and a shoe in your right (the objects are arbitrary - substitute where necessary). Now, look in the mirror. The hat will be on YOUR left and the shoe will be on YOUR right. There's no left to right swap. The reason that I capitalized "YOUR" is that I don't want you considering whether it's on the left or right of your REFLECTION. That's where the left to right inversion mistake comes from - we think the shoe is on the reflection's left. Normally, when we're facing another person, that person has rotated 180 degrees to face us. That puts her left on our right and vice versus. We mentally assume that's the case with the reflection in the mirror - that the reflection's right hand is on our left. However, the reflection wasn't rotated 180 degrees to face us. It was flipped front to back. The reflection's left hand is on our left, just as you would expect from something that has not been flipped left to right.

Ambulances have the word Ambulance flipped backwards, not flipped upside down.

This intuitively seems to be a major break of symmetry, why would a mirror have a horizontal preference?

It isn't the mirror with the preference, but the vehicle. To understand why, consider an ambulance facing you head on with the word "AMBULANCE" printed on it and the paths it could take to show up in your rear-view. It could move along a lateral half-circle or a vertical one (or something in-between). If it went along a vertical half-circle it would end up upside-down and behind you; since it goes along a lateral one it also gets laterally-flipped, but because cars have left to right mirror symmetry , you don't notice (except for the words ambulance).

We have a huge bias evolutionary bias towards that type of lateral symmetry (e.g. in animals, other humans) and to things moving along a lateral plane while maintaining vertical orientation.

I do not know whether this will help or confuse, but it is a different way to look at it:

Look into a mirror with your nose pointing North. Your left hand will point West, and your right hand will point East.

Your mirror image has a nose pointing South, a left hand pointing West (!) and a right hand pointing East (!)

You know it is his left hand pointing West because it moves when you move your left hand. Also, you may wear a ring or have distinctive features on one hand.

You know that left hand is pointing West because you can see the sun set in that direction.

So East and West stayed where they were. Also, up and down still are where they were; rain still falls on your head, not on your chin.

The directions that changed are North and South.

The effect of that is that, from the point of view of the mirror image, East and West seem to have switched sides. The mirror image is looking South, and yet, West is to its left and East to its right.

Feynman explains it best: http://www.youtube.com/watch?v=msN87y-iEx0&t=46

Face mirror. Hold book so you can read it. Now turn it to 'show it to the mirror'. The book is reversed right-left. Because you turned the book right-left.

Try it again - read the book. Now turn it to the mirror, but this time flip it top-bottom. Now the mirror is showing it reversed top-to-bottom, but the letters are still on the correct side (right on right; left on left).

So mirrors Don't reverse left-right. The do whatever you tell them to do.

And he's holding MTW's Gravitation tome in the photo!

BTW: Why do so few people pose with their favorite books (or holding any book) in profile photos, this was common when people had their portrats painted in the past. Has it become a turn-off?

Because it's not quite the same holding a Kindle displaying cover art up to your face.

Because most people don't read at all, or read BS (self-help books, how to succeed, cheap thrillers, 50 shades of gray...)

Have you read fifty shades? I find that people like to bag on stuff they've never actually experienced first hand.

Embarrassingly, I listened to a youtube video where a woman read the book aloud (I listened for about 30 pages). She ended up hitting herself in the head about a dozen times because the writing is that awful.

Fifty Shades is pornography for women. IMHO that's the only reason everyone is clutching their pearls about the bad writing. Your typical male pornography storylines/cinematography are much, much worse - and yet they are typically spared the high-brow literary scrutiny.

Yep, must be the patriarchy oppressin' again.

The reason that 50 Shades of Grey is subjected to highbrow literary scrutiny, is that it pretends to be highbrow literature. Male pornography usually makes no pretense, being more along the lines of logjammin' "ich bin here to fix your kable"

Not really. The reason 50 Shades gets its scrutiny is because it has enough exposure. There's some damn good male pornography (yes, it's literary; yes, I read it) out there, but it's not for sale at Barnes & Noble.

That's the patriarchy angle, if you actually want it.

In line with woah (minus the snark), male-focused porn is generally spared the scrutiny because it does not seek it. You won't read a review of Modern Whorefare (sadly a real thing) in the New York Times and you won't be able to pick it up at the grocery store. A difference in marketing and distribution amply explains the greater scrutiny, without resorting to implications of sexism.

Maybe because your "typical male pornography storyline/cinematography" is considered neither a "cultural phenomenon" nor "literature", and does not feature in newspapers and magazine reviews...

Male porn is just porn, but "50 shades of gray" is presented as it is a regular book...

Yes. Still crap.

Fifty Shades of Grey is just Jane Eyre rehashed - young virgin falls in love with rich powerful man. Same story, just a little bit more descriptive when it comes to the sex scenes.

The similarity ends on the plot summary level -- which is the most superficial part in literature.

It's also 20 levels below in writing quality, like a messed up straight 20,000 lines amateur PHP monstrosity compared to something written by Rich Hickey.

> And he's holding MTW's Gravitation tome in the photo!

I think it's because it's book with very large title font, which shows the mirror isn't reversing the text. It just also happens to be a favorite book of his (much better than if you saw "$AUTHOR_NAME" which is what you'd see for popular fiction, wouldn't you say?)

After knowing the Feynman puzzle, I think this is probably more correctly described as a reversing mirror.

If you look into a metal spoon, you'll see both X and Y reversed. So I'm wondering if you could make this mirror simply by only curving the horizontal part. That's kind of what it looks like in the photo. And if so, why didn't someone think of this early? It's quite obvious.

People thought of this before--just place two mirrors at a 90 degree angle. The guy's innovation is to make it very smooth.

I'm having a hard time visualizing this.

If I am standing directly in front of this mirror and slowly reach forward with my right hand until I touch the mirror, will my hand be touching its mirrored self?

I'm guessing that the only way you could touch the mirror image of your finger is by doing so in the middle of the mirror. If you move the finger out toward the shoulder, I'm betting your arm would stretch off the mirror away from your body. I could be totally wrong though.

Look at the cover of the book that the trick mirror is resting on.

It seems this mirror is especially shaped to project the ordinarily reflected image in a reversed way, thus putting an object near it surface will break the illusion and show skewed but normally reflected image.

Which seems to be what is going on with the book cover, in the picture in TFA.

Try this with a metal spoon. If you bring in your right hand, it will be on the left side, but extremely distorted.

It should be easy enough to simulate with a webcam.

For those interested in learning more about this, including the math behind the mirror's curve, check out Prof. Hicks's website: http://www.math.drexel.edu/~ahicks/

Relevant part (http://www.math.drexel.edu/~ahicks/papers/physics-today.pdf):

"As a second example, take the object plane and the image plane to once again both be at x = 1 and define T(1, u, v) ≡ (1, −αu, αv). As with the flat mirror of the previous example, the transformation scales the image plane, but be- cause of the relative minus sign in the y and z coordinates, the solution surface will not reverse an object as a conven- tional mirror does.

The relative minus sign also means, alas, that no exact solution surface exists. I obtained an approximate solution surface confined to the rectangular volume x = 34 ± 1 cm, y = z = 0 ± 3 cm, and constructed a prototype mirror. With α = 160, the mirror’s field of view is wide enough that I could see myself in the mirror when it was held at arm’s length, as shown in figure 1. To achieve the appropriate ray paths, the mirror is saddle-shaped."

So, this mirror only does its magic in a small volume; if you move your eye closer to the mirror or sideways, or even when you look at it with two eyes, the effect breaks.

From cursory reading, I even get the impression that this mirror is also tailored to the positions and shapes of the objects being photographed.

But as I said: cursory reading. Corrections welcome.

Feynmann's great mirror story [1]. Try and solve the puzzle before watching the whole clip.

[1] http://www.youtube.com/watch?v=msN87y-iEx0

How can he "invent" this? Hasn't it been known for years? It's the same principle as two mirrors at right angles, only instead of a corner, there is a gentle bend. I'm pretty sure I read about this in George Gamow's "One, Two, Three... Infinity".

This would be great for giving myself a haircut, but not so much for shaving. Probably because it is more left to right than back to front. I also shave more often and have built up my muscle memory more.

>The precise manipulations change the directions light rays are reflected off of the surface in a manner similar to changing the angles of millions of tiny facets on a flattened disco ball, but decreasing the size of each facet until a smooth surface is achieved.

Oh, so it's like a Fresnel lens, but it's a mirror instead of a lens. That's pretty clever. You get the non-reversing property of a curved mirror, but it's still flat.

Edit: Another way of thinking of it is that it's a normal map applied to a real-world object.

This is probably a small array of double mirror. If you look at the images you see slight loss of light (a bit darker) on his none-reversing mirror. Which is probably due to the gaps between the arrays behaving a bit like a scrim in shooting sets.

I think the invention will have some uses. For anyone who has tried to do anything that requires precision in front a mirror the reversing of the mirror is disorienting. I could even imagine mechanics who use a mirror to see what they are doing in tight spots be able to use it to coordinate their motion to put the ranch on the right nut when they are using a mirror.

No, it is not two mirrors. What Prof. Hicks made is a single mirror. The linked article explicitly says this.

Completely disagree with the OP's idea of putting this in a rear-view mirror. One benefit of a rear-view mirror is that things that are on the right side of the mirror are to the right of you, and vice versa. This mirror would break that, unless I'm misunderstanding it.

I'm interested in what applications this may have, if any really. I can see it working as a vanity mirror quite well, as then you can truly see yourself as other do. Aside from that?

The rear-view mirror mentioned by the article in passing is not a reversing mirror -- it's just supposed to be another example of this guy's expertise at fabricating mirrors, in this case to improve the field of view and to remove blind spots.

You can simulate one of these yourself by getting a wide flexible plastic mirror (dimensions along the lines of 1:4) and bending it to be concave until you look undistorted. Looking into this bent mirror, if you raise your right hand, the hand on your left side in the reflection will move.

Perhaps even easier is to secure two flat rigid mirrors at a right angle, like: /\

He's looking for uses beyond art galleries. I'm reminded of "The Man Who Mistook His Wife For A Hat" by Oliver Sachs.

(https://en.wikipedia.org/wiki/The_Man_Who_Mistook_His_Wife_f...)

Is this not just a reflective lens? I.e. a quadratic surface? If it is, I just assumed people didn't use them because non-reversing mirrors are actually rubbish. There's a reason cameras reverse the image in software but transmit/store the non-reversed image.

was hoping for more technical details. maybe they're still going through the patent application?

Here's the patent http://www.google.com/patents/US8180606 but it doesn't mean much to me.

This patent is not for the non-reversing mirror but for another type of mirror which was invented earlier.

Oh sorry, yep, you're right

according to the wikipedia article it's a surface of micro-mirrors that work in pairs. so it's basically a pair of mirrors scaled down and repeated.

http://en.wikipedia.org/wiki/Non-reversing_mirror

not very exciting really.

I imagine the hard part is actually optimizing that array of mirrors.

No, that is not what Prof. Hicks made. His mirror is a single curved piece.

Source: I am a former undergrad research assistant of his.

huh. thanks. so that's the second type listed at wikipedia? or am i confused? perhaps you or he could fix / clarify the wikipedia article? i guess i could but it feels odd to do it third-hand...

I wish I could give you a link to one of Prof. Hicks's published papers on the topic, but I'm having trouble finding one that isn't behind a paywall. This seems like a nice introduction: http://www.math.drexel.edu/~ahicks/papers/physics-today.pdf

If you have access to journals through a university, you could take a look at some of his papers yourself (His website is http://www.math.drexel.edu/~ahicks/). They're easy enough for a sophomore to recreate the results. (I know, because I did for an independent study with Prof. Hicks.)

The thing was around in 2009. This particular article is about an art exhibit that involves the mirror.

I can't wait until these are on the market. I could save so much money on haircuts. (Assuming the motor-confusion doesn't happen when using this mirror, because the image is not reversed)

A video of this mirror in action would've been very helpful.

Not exactly new: http://www.truemirror.com

No, that link is for 2 mirrors.

What Prof. Hicks made is a single curved mirror. That is new. That is why his results got published and he was awarded a patent for it.

If you had bothered to read the linked article, you'd already understand the distinction.

I agree that it is a neat implementation. I just meant that the idea of a non-reversing mirror is not new.

The curved mirror exhibits distortion at the edges, which doesn't seem to be a problem with the two-mirror approach.

Also the curved mirror doesn't present the image at full size, it appears smaller.

Breaking news.... HN abuzz with 2.5 year old story...

Hick's work was published via New Scientist (http://www.newscientist.com/gallery/dn16585-amazing-mirrors) in February of 2009.

I'd love to see a movie demonstrating what this looks like!

Here's a pic http://www.newscientist.com/gallery/dn16585-amazing-mirrors/...

That links also mentions that his amazing 45 deg. FOV side-view mirror is banned in the US, because curved mirrors are illegal in the US. What the hell? Doesn't congress have anything better to do than shoot us all in the foot, repeatedly?

a different implementation of the non-reversing mirror: http://www.truemirror.com/

bonus points: you can buy this one!

I want one.

This is pretty great!

Want.

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