Mastodawn

(Assuming we’re talking about the regular platonic shapes. I’ll accept no “look at this dagger I made with X sides!”.

I have my own answer and reasoning but I’m curious what others think).

Okay so I did some checking and even my assumption was wrong.

When you slice the points off of the shapes (at same distance from a vertex), the octahedron has less surface area, less volume, is less flat. However you want to measure it the octahedron point is more pointy. The cube's point is simply more blunt.

HOWEVER, the cube has more points (vertices) and is therefore more pointy.

Like I say, this was NOT what I was expecting and I may have made a mistake - but I don't think so.

The cube appears to be less pointy, probably because it is less pointy. (or rather, it has 8 points, they just aren't as sharp as an octahedron's 6 points.)

If we're talking about sharpness though, a cube's edges are absolutely sharper than an octahedron's. Be very careful around cubes!

Anyway, my conjecture (which has probably already been conjured by someone else but I am too lazy to check) is:

Polyhedra with fewer vertices have pointier vertices.

Polyhedra with fewer edges have sharper edges.

tl;dr: This is why you don't fuck with tetrahedra.

🇨🇦 TRH 🇨🇦Aug 13, 2023

@Sophie So "pointiness" means quantity of points and not sharpness of points. Got it. 😁
And octahedra are more dangerous.

YinYin Falcon Aug 13, 2023

@Sophie I reject defining pointy by quantity over quality 😤

words for that are spiky/prickly but even those would have trouble sticking to a cube, because it is right on the edge of blunt angles

Annette Aug 13, 2023

@Sophie Hmmm... how are you measuring the length? Along an edge? Would you get a different answer if you measured distance toward the centre of mass? I feel like the cube must be pointier because it subtends a smaller solid angle from one of its vertices.

@nettles that was my guess but those slices are made an equal distance toward the CoM - the height of the resulting pyramids is identical.

Annette Aug 13, 2023

@Sophie Whoa! That's pretty crazy!

Pete Alex Harris🦡🕸️🌲/∞🪐∫Aug 13, 2023

@Sophie
Maybe all the regular solids have the same total pointiness, but it's spread all over them and the ones with fewer points are pointier per vertex.

Miriam's on Medical Leave Aug 12, 2023

@Sophie Okay this is a fascinating and now I want to do so much geometry to figure it out.

@Sophie internal reasoning: which one do I less want thrown at me?

@skylerblue @Sophie The one with less mass?

@Mayor_of_Smartarse @Sophie Assuming they have the same density and volume, of course

@skylerblue @Sophie And distance from the target? Cubes aren't terribly aerodynamic, might be worth taking the chance on a miss... 😜

@Mayor_of_Smartarse @Sophie FINE Assuming all else is equal and the object *will* hit you…
*mumbles something about scientists*

@skylerblue @Sophie Well then, I'll take whatever is closest to a sphere rather than a brick 😂

@Mayor_of_Smartarse @Sophie Exactly! So at least we can agree that “Less Pointy” is synonymous with “More Closely approximates a Sphere”

@skylerblue @Mayor_of_Smartarse just want to step into the most pedantic conversation I’ve ever seen with:

A cube and an octahedron are duals so they are each equidistant to “spherical” depending on how you want to measure it.

@Sophie @Mayor_of_Smartarse Wait…
*smoke starts coming out of ears*

@Sophie @Mayor_of_Smartarse I'm sorry to spoil what I voted, but is it not Visually Evident that a Cube is Closer to a Sphere?

@skylerblue @Sophie Which would you rather step on in bare feet sneaking to the fridge at 1am? I'm going with 8 sides tyvm 😁

@skylerblue @Mayor_of_Smartarse “Visually Evident”

haha you trust your EYES to solve things? :p

@Sophie @skylerblue Ah, and here is me confusing octahedron and dodecahedron. (expletives)

@Sophie @skylerblue So, visually more spherical does not mean more spherical? Wow, my next D&D game is going to be a bloodbath...

@Mayor_of_Smartarse @Sophie Fuck it, math is a lie

@skylerblue @Sophie My point exactly.

Martijn Frazer Aug 12, 2023

@Sophie I voted yes, but also without realising what an "octahedron" actually is. Now that I'm looking at a picture of one, I think it's "no".

YinYin Falcon Aug 12, 2023

@Sophie no

sharper angles override more individual points - if you take either property to the extreme you are comparing a needle to a sphere

@Sophie If by "more pointy" you mean it has more apexes then no, but if you mean the apexes have smaller internal angles, then yes.
Interesting psychological experiment in any case. 😉

+>e Aug 12, 2023

@Sophie less and less pointy with more faces, as faces tend to infinity it turns into a sphere which are pointless

Zrephel Aug 12, 2023

@Sophie I have a perfectly reasonable argument that it is, which I am yet emotionally unconvinced by. An octahedron feels intuitively pointier to me. Perhaps it's that cubes can be packed neatly into a flat surface, and a grid doesn't feel pointy.

@Zrephel interesting argument, “if it’s space-filling, then there’s no space left to put my hand in and get poked”

Zrephel Aug 12, 2023

@Sophie I think also it's that I'm used to being cubes being axis aligned, in Minecraft-like worlds for example, and so a cube is made of "floors" and "walls". If you dual a cube to get an octahedron then the points stick out in the floor-and-wall directions. Likewise, an Egyptian pyramid feels pointier than a warehouse.

AndroidArts Aug 12, 2023

@Sophie This made me think of sea urchins, which are very pointy indeed – but wait – if you keep increasing their pointyness then they become fluffy instead at a certain hard to define...point. If you still count fluffy as pointy then cats are very pointy and that doesn't feel right in my heart.

@androidarts cats are definitely pointy, but it’s more because of claws/teeth than the fur. Anyway I think the distinction is whether a point bends. If it deforms easily it’s less pointy and more… flowy… maybe?

Farbs Aug 13, 2023

@Sophie I think a good measure of pointiness would probably be the proportion of volume outside the object in a sphere around the measured vertex.

Probably more useful would be that multiplied by two minus one, so that 0 is flat and concave is negative pointy.