Hacker News Reader

Comment by Out_of_Characte | Hacker News Reader

Negative area's can be the only practical solution. Imagine the worst case: a 10*10 room with a pillar in the middle. How would you calculate that?

View on HN

I don't see that as negative area. I see it as subtracting two areas. Both the room and the pillar have a positive area, none of them has negative lengths or areas.

Kids learn subtraction before they learn negative numbers - once you learn negative numbers, you know that addition and subtraction are almost interchangeable, but this is not necessarily intuitive to begin with.

Subtracting two areas is like adding two areas where one is negative…right?

Yes , but see my second paragraph. Just because something is true it doesn't mean it's intuitive.

That's interesting. To me it seems intuitive. It's a real area that can be drawn like any other. The sign is an operator describing what function to visualize, not a property of the measured area. So thinking of it in that way eliminates any need for the term "negative area."

But, intuition is subjective, so you may need to adjust the terminology to fit the visualization.

It is intuitive on some higher level, now that I know about signs, operators, negative numbers and all that. But when talking about visual information (i.e. "visual proof"), it isn't intuitive in that context.

In addition to that, for all I know there could be some pitfalls involved with negative areas which I'm not aware of. Even if there aren't any pitfalls, this isn't immediately obvious to someone who isn't familiar with the concept of negative area.

If I'm willing (or forced) to think in such abstract terms, I would much prefer an algebraic proof to this visual proof.

Your intuition is irrelevant if they’re mathematically equivalent.

There are no serious pitfalls with oriented areas. Adding them to your arsenal of geometric proof tools will greatly simplify many proofs. Not having such a concept makes ancient geometry books much more complicated than they need to be, often requiring lots of detailed case analysis where the separate cases are essentially the same, just oriented opposite ways.

It’s intuitive, just in more dimensions. People have different ideas and abilities to imagine/think dimensions, but on top of that we rarely train them to do that.

I think that build up to tensor fields should be in every school program. If you can’t think of a field, you’re mathematically disabled and too many basic ideas about real world are inaccessible to you. This limits the ability to vote on a set of topics and participate in non-local decisions that involve systemic understanding. Same for formal logic and statistics.

Once familiarized with that, you can easily start thinking of nonlinearly signed areas, complex areas and areas simultaneously positive and negative by an attribute.

The concept of negative area still feels like it'd get messy in a hurry. For a square pillar, the side lengths should be the same, suddenly giving you imaginary lengths just for the eventual area subtraction to work out. For a negative volume though, you need cubic roots of unity for the side lengths, throwing off your area calculations. Has anyone actually put together a system where the sort of concept you're describing is cohesive?

You're in for a nice trip, the concept is called Geometric Algebra:

https://youtu.be/60z_hpEAtD8?si=HHs_9m0IJ43nfI3S (~50m video)

TLDW: Yes, the concept is there, makes much more sense than a cross product (which is just an oriented area) and generalizes really nicely.

Alternatively, read: https://en.m.wikipedia.org/wiki/Bivector

Gotcha; it's less that my complaint doesn't apply, but more that it isn't relevant (i.e., "squares" and "cubes" aren't especially interesting constructs which need to coexist nicely, and if you relax that constraint then directional geometry can be very interesting). Does that sound right?

Reply on news.ycombinator.com