Comment by jerf | Hacker News Reader

Comment by jerf | Hacker News Reader

I like the problem statement, I like the use of a 2D representation, but the triangle symbol as drawn in that presentation is gargantuan. I think the symbol needs more work.

One thought that comes to mind is that I'm not convinced the bottom part of the triangle should be there. It implies a direction connection that I'm not sure exists. Removing that overlaps with ∧, logical and, though.

Another that comes to mind is that I can't think of another mathematical symbol that has such divergence in meaning depending on what is left blank like that. The closest I know of is integral, where you can leave the from and to parts blank for a symbolic integral, but that's still not like leaving those blank turns the integral into a differentiation (the opposite), depending. I'm sure there's something else somewhere up in math, too, but nothing your average student will hit.

Similarly, note that filling in all three corners of that symbol is actually an equation. I'm also not aware of any other symbols that constitute entire equations on their own. In fact hiding away an = symbol is probably a big strike against the idea as if anything standard math education underplays and abuses that most fundamental of symbols; let's not add to that. Again, somewhere up in higher maths than I've gotten to there may be symbols that constitute entire equations, but it won't be something most students see.

Also I think once that symbol is being shown with full expressions rather than cute little single-digit numbers or single-letter variables, it's going become very difficult to deal with.

I think there's something to this, though. I'm criticizing in the spirit of continuing to move forward. (I'm aggressively hostile to the idea that math notation is perfected and debating better alternatives is some sort of betrayal or something.) Personally I'd seek out a smaller, inline symbol that may visually reference a richer presentation (which may not be this literal triangle) for a nicer didactic experience, but doesn't literally draw it out in the formula.

> Similarly, note that filling in all three corners of that symbol is actually an equation. I'm also not aware of any other symbols that constitute entire equations on their own.

Multiplication and division can also be put into a triangle, such that filling in each corner constitutes an equation. See my comment [0] about the density equation. I suspect such triangles reflect some deeper relation of abstract algebra.

[0] https://news.ycombinator.com/reply?id=12138817&goto=threads%...

Well, sure, we can put whatever we like in explanatory diagrams. I'm not aware of recognized common mathematical symbols that constitute equations on their own, though. This video did not seem to be proposing a diagramming method, because if that's all it was there wouldn't be any controversy to speak of; it appeared to be proposing a new family of operators. That's what I was criticizing.

The triangle being an equation by itself is a bit of a bummer. The obvious followup to yours removing a line would be to remove two. Exponentiation would be:

    e
   /
  x

Logarithm would be:

  e
   \
    b

Root would be:

  x—b

Now a simple line may be too little. And it's not clear we should be using 3 different symbols ('/', '\', '—') either. But you get the idea. It's less gargantuan, and expressing the entire equation without the '=' sign, while possible, is so baroque nobody would do this except to make a point.

But what I like best with this idea is the possibility of replacing the line by some arbitrary symbol (including no symbol at all). That way, other triangular relations could be expressed without stepping on each other's namespace.

By the way, there is an obvious direction we haven't used yet:

  a
  |
  b

I'm not sure in which cases it could be useful, tough.

---

Now that I think of it, hiding the `=` sign may not be so bad in some cases. Relations don't have to be limited with the identity and orders (<, >, ≤, ≥). They don't have to be limited to relations with 2 variables either. Other relations would have to justify their usefulness to warrant a special symbol of their own however.

> One thought that comes to mind is that I'm not convinced the bottom part of the triangle should be there. It implies a direction connection that I'm not sure exists.

The operator in question is actually quite common.

> Buying stocks. Suppose you buy $1000 worth of stocks each month, no matter the price (dollar cost averaging). You pay $25/share in Jan, $30/share in Feb, and $35/share in March. What was the average price paid? It is 3 / (1/25 + 1/30 + 1/35) = $29.43 (since you bought more at the lower price, and less at the more expensive one). And you have $3000 / 29.43 = 101.94 shares. The “workload” is a bit abstract — it’s turning dollars into shares. Some months use more dollars to buy a share than others, and in this case a high rate is bad.

https://betterexplained.com/articles/how-to-analyze-data-usi... (harmonic averages section)

The concept describes various processes which contribute towards identical workloads at different speeds (or rates).

I'm not sure how to relate what you said to my post? Wrong reply button?

> One thought that comes to mind is that I'm not convinced the bottom part of the triangle should be there. It implies a direction connection that I'm not sure exists.

I should have included this quote to begin with. I'll put it in now.

In any case. A direct connection does exist. If you follow the link, there are other examples where the (+) operator would be useful besides parallel resistance.

Now I see. Thanks.

I still think it implies a degree of symmetry that will confuse students, though, and that the notation can be further improved from what was proposed there. The operation in question is not trilaterally symmetric, and using a trilaterally symmetric symbol is probably misleading. The root symbol is arbitrary, but at least it doesn't promise nonexistent symmetries.

The root symbol (as a multi-valued function) returns the roots of unity (multiplied by a scalar). Which is like, the posterchild of imaginary symmetries. </pedantry>

More seriously. The selling point of the triangle operator is that it highlights isomorphisms between the three more traditional operators as a cyclic group. Which is a symmetry, just not the commutativity that students might naively expect. So I suppose it's a double-edged sword. I agree that it sucks as an operator. Nonetheless, the video could help a lot of students.

I am sympathetic to the aims here. I think it makes a lot of sense to represent the relation (a,b,c) where a^b = c in a unified way. Of course, it may also make sense to represent other ternary relations in similar ways, too. You could put something inside the triangle to represent which relation is intended.

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