The Renewal Equation

A math odyssey

Experimental Mathematics

I spent most of my study time today conducting experiments with shadows, inspired by the trigonometry problems I discussed yesterday. In the two OpenStax exercises, the only measurement that could conceivably be related to the angle of the spotlight was the height of the shadow. Yet it seemed to me that the height of the shadow was determined by the other measurements. Since those were clearly independent of the angle of the spotlight, I suspected that the height of the shadow was as well.

I decided to test this experimentally. As shown in the photos below, I glued a small upright representing a human to the floor of a box and cut an aperture at floor level in front of it. I then shone a light through the aperture to cast the upright’s shadow on the opposite wall of the box.

Upstage
Downstage

The tools I had for my experiments were imperfect, but after multiple trials with different light sources, I am nearly certain that the height of the shadow cast by the upright is independent of the angle of the light. Below are two photos. The first was taken with the light at a low angle and the second with it at a high angle. As you can see, the height of the shadow is unchanged.

Low Angle Shadow
High Angle Shadow

The only thing that changed the shadow was moving the light source. This led to some difficulty, since with the tools I had, it was hard to change the angle of the light without also changing its location. (For instance, in this video, in which I tilt a phone light from zero to about 45 degrees and back, I attribute the slight changes in the shadow to changes in the distance of the light from the aperture.) Still, this gives insight into how the problem I brought up at the end of yesterday’s post differs from the ones I was confused by. The “angle” of the sun depends on its position, while the angle of the spotlight does not.

Should I contact OpenStax about this issue? Let me know what you think.

Responses

  1. Tim McL

    I adore the experimentation!

    I think, however, that the problem isn’t very clear and that you’re making it more complicated than they intend. When they say “spotlight,” I think they mean a very narrow beam. (That’s what’s spotty about it, and what’s not spotty about your experimental setup.) What if you replace “spotlight” with “laser?” The laser is mounted on the ground and shines up just touching the person’s head before burning a hole in the wall. The shadow is everything on the wall below the smoking hole. Then you really have more info than you need to determine the laser’s angle. If it’s foggy and the laser beam scatters some, then you really get a shadow.

    If that’s how they want us to read the problem, then it makes sense and it easy. Otherwise, with the interpretation in your amazing experiment, then I agree that the problem doesn’t make sense.

    What do you think? Is it possible that this is what they had in mind?

    Reply
    1. Olly

      Yes, I think the problem you propose is probably the one students are meant to do. Or, at least, its answer is the one they were meant to find. It really isn’t the problem set, though. The problem describes a six-food shadow, which could only be made by a fairly broad beam of light.

      Reply
      1. Tim McL

        Ah. A six-food shadow makes the problem much tastier than I thought it was at first. 🍩🍭🍥🥨🍕🥚 👤👤
        (What are the most mathematically shaped food emojis? 😉

        Reply
        1. Olly

          I’d say the donut and the pretzel.

          Reply
  2. Tim McL

    Another possibility is that they want a comical (well, conical. Thanks, phone) beam whose top just touches the person’s head and whose bottom just touches their feet. They’re asking you for the angle between the ground and the center of the beam. I don’t think this is half the angle to their head, but it shouldn’t be too hard to get. Maybe that’s the most trigonometrically reasonable question? You could always do that one, and then go on with your life. It’s a lousy problem, anyway.

    Reply
    1. Olly

      This sounds potentially interesting. I have gone on with my life, but maybe I’ll circle back to this at some point.

      Reply
  3. Lucian S

    Yeah, I assume what they are actually after is ‘what is the angle formed between the ground and the light that forms the very top of the shadow’. Which is awkward, and you should definitely contact OpenStax!

    Reply
    1. Olly

      Perhaps I will link them to the two blog posts. I’m pretty sure these exercises were not meant to be the type that generate discussion.

      Reply
  4. yerpa

    I don’t think it needs to be a laser, though that could clarify thinking about it. I think ” at what angle is the light” really meant “what is the angle from the light on the ground to the top of the shadow (or the top of the subject’s head)”. A wide light that radiates at a sufficient degree from the source (not a laser)will cast the whole shadow. For the sun that “sufficient degree” can be a very tiny fraction of a degree, because it is so far away. For Olly’s box* the source isn’t very far away, so it has to be something of a floodlight to make the whole shadow, but the only rays of that light that concern us are those going just over or nor quite over the figure’s “head” since those infinitely-close-in-angle rays define the top of the shadow. Then we are back to Tim’s Laser (not to be confused with Occam’s razor).
    The angle at which the floodlight is shone is irrelevant, as long as the diverging beam is sufficient wide to span the whole figure (or we only care about the “head” shadow); we remain concerned only with that (or those, if an infinitely small gap bestows plurality).
    The question was simply poorly worded, a problem familiar in mathematical thinking, that has led to the preference for equations.

    *which, in contrast to Schoedingers, does not contain a cat, but an unarguably dead hunk of cardboard with a human figure drawn on it

    Reply
    1. Olly

      Adding a cat to my box would probably have made the experiment more memorable, but still not as memorable as Schrödinger’s.

      Your interpretation of the problem is a good one. I will probably propose “what is the angle from the light on the ground to the top of the shadow” as a rewording if I contact OpenStax.

      Reply

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