Wednesday, June 20, 2007

The Dimensions of Shadows


In honor of the recent Father's Day, I'm posting about a debate I once had in the car with my dad. We were talking about shadows and dimensions. Okay, so this is more math, but if you like physics, you must at least have some appreciation for the power of math.

Dad believes that a four-dimensional object could cast a three-dimensional shadow. I was convinced that all shadows must be two dimensional.

In my mind, a shadow is just a two-dimensional map of places where light has passed by or been obstructed by an object. So it must be two dimensional. Wikipedia says that I'm just talking about the cross section of a shadow -- they're all essentially three-dimensional.

It's hard to imagine a four-dimensional object, though. If I was better at math, maybe I could say something about how photons would interact with it, but I don't have much of a conceptual idea. Some folks at Union College of New York who are better at math made short animations of the shadows of rotating cubes (3D) and hypercubes (4D, projected in 2D for the picture).

In the animation, each face of the hypercube casts a three-dimensional shadow. So I guess the math supports my dad. It's really hard to argue with math. But if you have an opinion on the matter, I'd love to read it.

Photo Credits: Wikipedia

19 comments:

  1. I don't think I played with legos enough as a kid.

    ReplyDelete
  2. Actually, I spent a lot more time with Barbies than with Legos as a kid. But that's another story.

    ReplyDelete
  3. I think the shadow is generally the same dimensions as the space you are in. In 3 dimensions, the shadow is also three dimensional and then the surface (e.g., a wall or side walk) provides a 2 Dimensional cross section of the shadow.

    Same thing should happen in 4D. The shadow will be 4D, and if you take a projection on a 3D surface, it would be a 3D object or it could be 2D if you choose to take a cross section on a wall which is 2D.

    ReplyDelete
  4. That makes a lot of sense. But man, I'm so used to 2D surfaces, it's hard to wrap my mind around one that's three dimensional. I can appreciate what math says is possible, but I am fundamentally locked in a 3D reality.

    ReplyDelete
  5. Actually shadows are three dimensional. During a Solaw Eclipse, for example, when the moon blocks out the sun it casts a shadow on Earth. If you were to look at this from a perfect right angle, a side view, you would see the "umbra". The Umbra is like the "cut" or void in between the light, the space where the light is absent. This is shadow occupying three dimensional space.

    When the shadow hits a surface, the surface displays a cross section of the shadow. The cross section is 2 dimensional, but the shadow itself is 3dimensional.

    If a 4-D hypercube were hit with visible light, it would also cast a 4 dimensional shadow, but the cross sections of the shadow would be three dimensional.


    If it's too difficult to imagine, just think of it this way:

    0D= point
    1D= line
    2D= square
    3D= cube
    4D= tesserect (hypercube)


    The points are the surface of the line. The line is the surface of a square. The square is a surface of a cube. And a cube would be the surface of a tesserect.

    ReplyDelete
  6. Dear Sir,

    I think all shadows are in 3 dimension because we are living in a 3 deminsional world. And I don't think it's possible to create a 4th dimensional object. Our eyes can only see in 3 dimension. However, I have one question. Is our reflection in a mirror in 3D ? Because we can't feel our refection. But maybe it is also in 3D because it is similiar to a shadow.

    And I want to say something which is also a wish. Why won't God himself answer all of our questions so that we won't have to think too hard to come to an answer?

    ReplyDelete
    Replies
    1. Because God wants us to think. He wants us to question things and to never stop learning.

      Delete
  7. Dear Sir,

    I made a mistake by assuming all shadows are in 3D. A fog shadow is in 3D. A shadow on a wall must be in 2 diminsional because you can't look at it behind it. And a reflection in a mirror is in 2D because you can't see it behind it.
    You might as well ignore my previous comment I made at 2:47 AM.

    And I don't think God will bother to answer any of our questions. It's as if he doesn't exist.

    ReplyDelete
  8. even in a well lit room a shadow cast exists in 3 dimensions. if the shadow is dark enough to project an image on a flat surface (which you view as a 2d image) it is limiting ALL light in between point A (light source) and point b (flat surface)

    Since this is a constant it must be true to say the dim space between point A, and point B, is 3 dimensional.

    ReplyDelete
  9. Does a shadow have thickness? If so, what controls its thickness? Are all shadows equally thick, regardless of how tall or wide they are?

    And what about shadows within shadows that are created when light from two sources hits an object from different angles? Is it "thicker" than a shadow cast by a single light source?

    Shadows may represent the distorted height and width of the object that created it, but how can it have thickness? If it does, how can its thickness be measured?

    Maybe the "umbra", which represents a void where light is absent is a fourth dimension.

    ReplyDelete
  10. 4 Dimensional shapes CAN cast 3 dimensional shadows. Maybe not on the ground... but through smoke..

    ReplyDelete
  11. i can only barely imagine these. when i think of the first dimension i cant realy understand it completely becaus if you think about it, any surface must have something holding it together right? i think a point or a line is still 3 dimensional cause if you could zoom in to it well in my mind it gets weird becaus it turns out to be 3D..i mean atoms are 3D so.. and a shadow, well if you try n zoom into that, could u see any side to it? and does a 1 dimensional "something" have anything inside of it, or does it have a side or a top and bottom? i just cant imagine it disapearing when it turns to the side, their must be something their. and if their was it wouldnt be 1D..someone pleaze comment becaus this is bothering me.

    ReplyDelete
  12. Ok someone PLEAZE help me here..how many dimensions can light have? Or in this case how many can no light have? "a shadow" is NO light. Its a visual description of how much light isnt their. and i dont know much about dimension the more i think about it..What does dimension describe exactly? Isnt it just what we "find out" or precieve in any frequency, object,visual thing,or waves? i mean i heard 1D is just a point? or line..and when i visualize it, i cant except it as an exizting thing becaus a point or line must have a top and botom or a side. becaus if you turn a 1D object to the side it would be 3d..so a 1 or 2d object would have to be "like" a shadow..right? but then it wouldnt exist without the 3D..omg..whatever i have to stop thinking about all these or my head will explode..someone please explain!

    ReplyDelete
    Replies
    1. love your idea, heres one though, a shadow is three dimensions. It has a top and bottom, front and back, and a left and right, just because you cant see it dont mean its not there, and it has a weight. you just cant measure it, (yet) a shadow has mass it has to, which in return makes it matter, all matter that exist in our universe.... wait could this be the dark matter everyone is looking in space for. maybe a shadow is the fourth dimension, its been here all the time.

      Delete
  13. Shadow depth = planks constant(h) / h = 1Dimension

    ReplyDelete
  14. the fourth dimension is the potential of timelessness and spaitalessness. in a way it is umbra.

    ReplyDelete
  15. *Sees fourth dimension in previous comment. Sips coffee and prepares to rant.*

    I try to imagine it like shadows of everyday objects I see. Just the outline of them. Every single detail on the 3-dimensional surface of them, in varying degrees of "shadowiness" depending on the intensity and source of light.

    But here's the first problem we run into. There's another dimension of measurement at play here that we just simply cannot observe, therefor whatever it is that one sees is not accurate of everything that is actually taking place.

    I believe a 4-dimensional object's shadow would look somewhat similar to a kind of "shadow version" of our 3-dimensional objects we interat with everyday. The difference would be that we could move through said shadows as they would be freestanding, unbound by floors or walls. Think intricate and sophisticated hologram projections resulting from the absence or blocking of something rather than a clever projection of "things."

    'What parameter is this whole fourth dimension thing, really?' question is still nagging. Would shadow cast of such an object even matter if it's a simple light source like the one's we know? Do light and shadow operate by entirely different physics in 4-D? If you buy into the whole time travel bit, then would we be able to interpret the future, past or simulatneous existences' "shadows" being cast over our 3-D world?

    *Sips coffee to keep head from exploding.*

    Would we be able to "feel" these shadows, moreso than the way we feel current shadows? If I stand in the shadow of a tree while the sun's out, I'll feel the temperature change versus if I were to stand directly in sunlight. So, if a 3-D body (us) under a 2-D shadow (tree) feels change on a scale of one first degree (temperature), does a 3-D body "inside," as it were, a 4-D shadow feel change on a scale of more than one degree?

    To put it simpler: in addition to feeling a temperature change, what other effects are there to be felt when cast in a four dimensional shadow?

    In regards to the super shapes: I'm sure a model could be built that does a three dimensional rendition of the tesseract morphing inward and outward. It would simply be a representation of the 3-D animation on a 2-D screen, not true 4-D. Something that can expand and constrict autonamously basically. While the super shapes like the hypercube and such are fascinating, I believe there is something fundamentally missing from the discussion about 4-D objects that goes far beyond their perplexing shapes and designs.

    ReplyDelete
  16. Look at it like this. If I can take a point (1 dimension) and somehow turn it back on it self, for example, drawing a line or a shape, somehow connecting it again to where it is more than just a blip, it would then have height and width (2 dimensions). Knowing this, I could repeat this process infinitely - rearrange what already exists in order to grow by an observable dimension, thus increasing perspective and potential.

    Fold a sheet of paper (2 dimensions) till it's a cube (3 dimensions). Now that one area near the top right corner of the paper has attacthed to another part of itself, and in the process achieved a higher dimensional value. (See where I'm going with this?)

    But in the fourth dimension, transforming 3-D shapes, objects, creatures in such a manner becomes.... troublesome. Afterall, we'll talking manipulating something a bit more harder to control than mere paper. Something like that would require power a bit more potent than simple ingenuity an imagination.

    Or... would it?

    Life, no matter how small or complex, is only supported at the third dimension of existance. (Not too many sentient geometrical shapes out there. Sacred geometry, on the other hand...) It stands to reason that dimensions beyond this could also support life. Yes, a line or a dot are technically 2-D but ironically we can only represent these things using three dimensions. If you press down on a sheet of paper to draw a dot, you've left a piece of ink that has a measurement of not only how wide and long it is, but even how far from the page's surface to the top of that droplet of ink. Three dimensions. Always.

    Back to the idea of the paper to a cube - it was in the third dimension all along, but for the sake of this example let's just say it was only 2-D. Many theorist look to time travel as a reasonable eplanation as to what that dimensional increase would look like past the third dimension. If that is the case, again, what would the SHADOW of something from the fourth dimension look like?

    Perhaps the answer is as simple as finding out how 4-D light works?

    At any rate, the process required for the continuing the aforementioned example of dimensional increase no doubt exceeds our current understanding, however I am certain the shadow of such a thing is equally beyond what we've currently experienced as well.

    ReplyDelete
  17. Rotation in four-dimensional space

    https://youtu.be/R1rzI0Y_d3I
    The 5-cell is an analog of the tetrahedron.

    https://youtu.be/BjvdrhK8yws
    Tesseract is a four-dimensional hypercube - an analog of a cube.

    https://youtu.be/Pa0c7M4lZv0
    The 16-cell is an analog of the octahedron.

    https://youtu.be/np0ZxC1wXqc
    The 24-cell is one of the regular polytope.

    https://youtu.be/T01qw0_qitI
    A hypersphere is a hypersurface in an n-dimensional Euclidean space formed by points equidistant from a given point, called the center of the sphere.

    ReplyDelete