My main question is in Scenario 2. I don't know how to answer this main question.
Scenario 1 asks a similar question that I am able to answer, that led me to think of my main question.
Scenarios 3 and 4 ask similar questions to my main question, that I attempted to try to help me answer my main question.
Scenario One -- laying down a flexible piece of paper onto an sphere, and tracing the resulting shape
Suppose I have a 1cm x 1cm piece of paper, and a sphere that is somewhat larger (for example, a sphere about the size of an orange).
Suppose I lay the square piece of paper down on the orange, and I flatten it as much as I can. It is impossible to do this in a way where all of the piece of paper will touch the orange; some parts might be crumpled "into the air" where air is in between the paper and the orange; and some parts might be lying close to the orange, but paper is in between that part and the orange.
Suppose I then use a maker to trace a shape onto the orange, by tracing around where the edges the piece of paper touch the orange.
- A question: how does the surface area of this traced-out shape compare with the area of the square piece of paper?
- My answer: Intuitively, I can visualize that the surface area of the shape traced onto the orange will be less than the area of the square piece of paper, because only some points of paper are directly touching the orange; there are other "extra" points of paper that are in the air or lying on top of other points of paper. These "extra" points of paper are not contributing to the shape that is traced out, therefore the shape that is traced out must be less than the area of the square piece of paper.
Scenario Two -- rolling a rigid piece of metal onto a sphere, and tracing the resulting shape
Suppose, now, that I have a rigid square piece of material (such as a piece of metal) that cannot be bent, that is 1cm x 1cm, and a sphere that is about the size of an orange.
Suppose I use this piece of metal to trace a shape onto the orange in the following way:
- I start by placing the centre point of the square onto the orange, such that it's the only point touching the orange.
- Then I smoothly roll the square piece of metal along the orange, making a line from the centre of the square piece of metal, to the midpoint of one the edges of the piece of metal. I then mark a dot onto the orange, with a marker, where the midpoint of the edge of the square touches the orange. I then undo this rolling motion, so that the centre point square piece of metal is once again touching the orange.
- Similarly, I mark the midpoints of the other three edges of the square (ie, starting from the centre, and rolling it to those midpoints). And similarly, I mark the corners of the square. And then, I mark many, many other points along the edge of the square.
- Finally, I can join all the points marked with some sort of smooth (possibly curved?) line.
Main Question: A shape now is "traced" onto the orange. Is its surface area less than, equal, or greater than 1 square centimetre?
I have trouble even making a guess! (I welcome any observations that may help me visualize certain properties of this scenario, even if they don't provide a mathematically rigorous explanation. I especially welcome explanations that use high-school (or earlier) level mathematics.)
Indeed, I can't even make a guess about what shape is traced! Are the corners of the traced shaped right angles? If looking directly down at the shape (ie from a bird's-eye-view, as if your eye, the centre of the shape, and the centre of the sphere, made a straight line), would the shape look like a square? I can't visualize this!
Similar Questions I asked myself, as attempts to gain an understanding of the Main Question
Scenario Three -- tracing a rigid square onto a cylinder
Suppose a trace a shape onto a cylinder in the following (obvious) way:
- I lay a cylinder flat on a table (ie, so one of it's circular faces is resting on the table).
- I then place one edge of the square so that it is perpendicular to the table, against the side of the cylinder, and I draw a line along that edge, with marker, onto the cylinder.
- Then I smoothly roll the square towards the other vertical edge of the square. As I smoothly roll the square, the horizontal edges of the square will "roll along" the cylinder. As they do so, I mark the places where they contact the cylinder.
- Finally, I draw a vertical line onto the cylinder, when I reach the square's other vertical edge.
I am somewhat confident that the surface area of the resulting shape would equal one square centimetre:
- It seems that (as I'm rolling the square along the cylinder) if I take any two points on the square that are vertical to each other, the distance between these two points on the square will be equal to the distance between the points where they each touch the cylinder (as the square is being rolled onto the cylinder).
- Also, suppose I consider any two points that are horizontal to each other on the square, and I mark where these points touch the cylinder (as the square is being rolled), using purple ink on the cylinder. If I measure the distance between the two points on the square, and if I measure the distance an ant would walk along the curved surface of the cylinder between the two points in purple ink that I marked on the cylinder, my guess is that these two distances would be the same.
Considering this, it feels like all points of the square will be "placed" onto the cylinder's surface without any distortion/warping/stretching. That's why I am making a guess that the surface area of the resulting shape traced onto the cylinder will equal the surface area of the square piece of metal.
Scenario 4 -- Tracing a square onto a cone
I am not confident in my guess here. My guess is that the shape traced onto the curved part of the cone will have a surface area equal to the area of the square.
However, I have trouble even visualizing the tracing process.
Suppose I start with the cone resting on the table (ie, with the circular face of the cone flat on the table). I note that I can choose a line on the curved surface of the cone, that runs from the tip of the cone down to any point on the edge of its circular face. I then start by placing the square onto the cone, such that the midpoint of two opposite sides of the square align with the "line" on the cone that I chose earlier. I can then do an obvious "rolling" motion of the square, once going right, and once going left, to trace out a shape onto the cylinder.
But I am unsure if I can visualize the shape created:
- For example, when rolling the square, starting with the starting placement described above, and then rolling right, there will be a line traced out (onto the cone) by the upper horizontal edge of the square, and a line traced out by the lower horizontal edge of the square. Will these lines that are traced out be parallel to the table? My intuition is saying "no"; my guess is that that the lines will slant downwards towards the table, as you roll the square from the middle of the square to its right edge .. but I'm not sure.
- Similarly, the vertical(ish) line created by the vertical right edge of the square .. will it be "vertical" (ie, aligned with a line that runs from the tip of the cone, down to a point the edge of the cone's circular face)? I cannot visualize this well enough to even make a guess.