Why slicing a cone gives an ellipse

You can also use the same trick to explain why cutting a cylinder at an angle produces an

ellipse

, and if you're comfortable with the statement that projecting a shape from one plane to another inclined plane has the effect of simply extending that shape, this too explains why the definition of an

ellipse

as an extended circle is similar to the other two definitions. More tasks. So why do I think this proof represents mathematics itself so well? That if you had to show just one thing to explain to someone who is not passionate about mathematics why you love the subject, why they would be a good candidate.

The obvious reason is that it is objective and beautiful without needing much background, but more than that, it reflects a common characteristic of mathematics which is that sometimes there is no single, fundamental way to define something, and what matters most is showing it. . equivalences. Even more than that, the guide itself includes a key moment of creative construction, aggregating the two fields, while most of it leaves room for a systematic and principled approach. I think this kind of creative construction is one of the most thought-provoking aspects of mathematical discovery, and it's understandable that one might wonder where such an idea came from.

In fact, speaking of this particular evidence, this is what Paul Lockhart says in the analogy. I agree with that, but I think we can at least say something more about it. While that's clever, perhaps we can discuss how someone immersed in a number of other engineering problems might be particularly willing to consider adding these specific areas. First, one of the common techniques in geometry is to relate one length to another, and in this problem, you know from the beginning that being able to relate these two lengths with the foci to some other lengths, especially those that are aligned, is going to be something Well, even though you're just getting started.

You don't even know where the focus points are. Even if it's not clear exactly how it's done, throwing balls in the photo isn't crazy. Again, if you have built a relationship with geometry through practice, you will be very aware of how the connection between one length and another happens all the time when there are circles and spheres in the image, because it goes right to the stamp of geometry. What does it even mean to be a circle or a ball? Obviously this is a specific example, but what I want to point out is that flashes of ingenuity can often be seen not as inexplicable miracles, but as remnants of experience.

When you do that, the idea of ​​genius goes from magical to actively inspiring.