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While studying the history of photography, I became interested in this property of daguerreotype. As stated by Wikipedia:

"The image is on a mirror-like silver surface, normally kept under glass, and will appear either positive or negative, depending on the angle at which it is viewed, how it is lit and whether a light or dark background is being reflected in the metal."

I don't understand how this is possible.

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It's possible because the tonal range of the image is rendered primarily as a range of diffusion of reflectivity. The highly-exposed areas, what we would normally think of as "lighter", are rendered more diffuse through added fine texture, making them somewhat more like paper - the brightness changes in a fairly predictable way depending on the overall lighting.

The less exposed areas, however, retain a more mirror-like finish. If you observe objects made of shiny metal, you'll notice that they can appear dark or bright depending on exactly what they're reflecting to your eye at the moment, which can depend on the angle of viewing, etc.

So, essentially, you have an image where the normally-darker areas can vary in appearace over a huge dynamic range depending on viewing conditions, while the normally-lighter areas vary much less from a "middle" range. This can sometimes result in the "dark" shiny areas appearing lighter than "light" diffuse areas, if they are reflecting a light source towards your eyes, resulting in a "negative" effect.

UPDATE: I took a couple of shots illustrating this, since it's easier to understand visually than to explain:

enter image description here enter image description here

The difference was created simply by moving the camera a short distance to the right, away from the ambient light source on the left; the exposure was decreased to approximate the eye's adjustment to the change in brightness.

(Original photo is c.1850, and should be well within the public domain in most countries. Any rights to the derivative works are hereby released under CC0).

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As you know, the Daguerreotype is actually a negative image on a mirror background. You can see the same effect if you play around with old black & white negatives. Perhaps you can find some, maybe your grandmother or grandfather has some in a shoebox. Place one or two on a hand mirror and look at them under different lighting conditions. Now place them on a shiny black surface and surprise, your negatives will appear as positive. Maybe you would like to know that the tintype came next. These were made by coating metal plates with a high gloss black shellac known as “Japan Black”. Then the plate was coated with a light sensitive emulsion.

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