6-sided is possible if you do the pointy outside first, then the hexagon in the middle. 9 Should be possible in some similar way, too. It highly depends on which parts of the star you think are needed for the star, as you are not really consistent there. Is it just the pointy things? Pointy things and one multisided area in the middle? These two always work no matter how many spikes you want on your star. Additional things might lead to problems, but probably not. It could stop working if you require all lines to be drawn from one edge to another, like in the case of the 6 sided one.

Najda

United States3765 Posts

June 13 2014 04:55 GMT

#2102

On June 13 2014 13:49 Simberto wrote:
6-sided is possible if you do the pointy outside first, then the hexagon in the middle. 9 Should be possible in some similar way, too.

True, but I should have been specific in saying you can only draw lines from point to point. Also each point has to have the same angle. I can draw a lopsided 6 pointed star like this:
[image loading]

but it doesn't meet the angle requirement.

Simberto

Germany11800 Posts

June 13 2014 04:59 GMT

#2103

Well, in that case anything with an even number of points larger than 4 does not work. The 8 sided one you painted is kind of noneven too. What kind of symmetry do you require exactly?

Najda

United States3765 Posts

June 13 2014 05:03 GMT

#2104

On June 13 2014 13:59 Simberto wrote:
Well, in that case anything with an even number of points larger than 4 does not work. The 8 sided one you painted is kind of noneven too.

That was my initial instinct but I can't get 9 to work, but 8 fits my definition of rotational symmetry, each point being the same size/internal angle. If it was a physical object you could roll it and it would maintain symmetry any time it's resting on two points.

Simberto

Germany11800 Posts

June 13 2014 05:14 GMT

#2105

Hm, that is a lot harder to math. Basically, if you go with "you need rotational symmetry in the order of points of the star", i think it would be not too hard to math out. I think that might work only for primes, or something along those lines.

But if you only go with "All points have to have the same angle", it gets a lot harder.

Edit: 9 works by always going to the point 2 further.
10 by going 3 further.

All of those have n degrees of rotational symmetry too, so the idea with primes was obviously nonsense.

Najda

United States3765 Posts

June 13 2014 05:18 GMT

#2106

I think you could represent it mathematically by calculating the interior angle for an n-pointed star and requiring that each new line has to be drawn to a point with no pre-existing lines except for the final line to close the circuit (the nth line). If no one has mathed it when I wake up tomorrow I'll put more thought into it, but sleep for me now.

Thieving Magpie

United States6752 Posts

June 13 2014 05:28 GMT

#2107

On June 13 2014 13:46 Najda wrote:
[image loading]

For the 5, 7, and 8 pointed stars I drew, you can do it without lifting up the marker. Is there a method to draw the 6 pointed star without lifting up your marker? If not, is there a mathematical reason why not? Similarly I don't think 9 is possible (though you can draw it by stacking three triangles) and I didn't try 10, maybe it has to do with multiples of 3?

The 6 point star is ridiculously easy to do without lifting.

Before you finish the 3rd side of the triangle, start the 2nd triangle until you get back to the point you had stopped with the first triangle and then finish it.

It is only impossible to do if you *have* to complete a line in one stroke.

Simberto

Germany11800 Posts

June 13 2014 05:35 GMT

#2108

An n-pointed star is not really rigidly defined, though.

After some thinking, i have come up with this solution: Basically, what you need to draw a good star is an integer x that fulfills the following conditions:

x<n-1 (If x=n-1, you are drawing a polygon, which i assume do not count as stars because otherwise this would be trivial)
x>1 (Same as above, x=1 leads to polygon)
x and n do not share any prime factors.

If at least one number like that exists, you can draw a star without taking your pen of the paper by always drawing a line to the point x further from where you are, and since there are no common prime factors, you will have to fill up all points and thus finish a star. I am pretty sure that if no x exists that fulfills those conditions, you can not draw a star that has n degrees of rotational symmetry. If more then one x exist, you can draw multiple different stars.

This means that:
1,2,3,4,6 do not work.

No idea if there are other larger numbers that do not allow continuous stardrawing, but i doubt it. Right now i can not proof that that is the case, though.

Najda

United States3765 Posts

June 13 2014 05:40 GMT

#2109

On June 13 2014 14:14 Simberto wrote:
Hm, that is a lot harder to math. Basically, if you go with "you need rotational symmetry in the order of points of the star", i think it would be not too hard to math out. I think that might work only for primes, or something along those lines.

But if you only go with "All points have to have the same angle", it gets a lot harder.

Edit: 9 works by always going to the point 2 further.
10 by going 3 further.

All of those have n degrees of rotational symmetry too, so the idea with primes was obviously nonsense.

9 and 10 don't meet the one continuous point to point line requirement though, and I am willing to dismiss 8 on account of the points all not being rotationally identicle (just mirrored). Primes is an interesting theory, I'd be interested to see why the math works like that if it's the case.

Simberto

Germany11800 Posts

June 13 2014 05:42 GMT

#2110

I'd suggest reading my new post just above yours then. Primes are nonsense as it turns out.

And 9 and 10 do work out. Picture inc once i am done paint-ing.

First one is using x=2. Second one is using x=5. Third one uses x=3.

Now, these might look shitty. But the reason for that is that i am too lazy to do it properly. If you do it properly, they are wonderful pretty stars that are drawn without taking the pen of the paper. Also they do have the same angles on each point, and n degrees of rotational symmetry. This way of construction should work for pretty much any larger number as long as you can find an x that fits what i described above. Most numbers probably have more than one x like that, but i can't completely exclude that larger numbers exists where there is none. I'm gonna think on how i can exclude that.

Ok, now i am rather sure that no number larger then 6 exists for which this can not be done. Because that would require that number to be dividable by all primes smaller than itself. I can not rigorously proof that no larger number with that property exists, but considering i am a physicist and not a mathematician, the fact that if such a number exists, it is probably so large that you can't reasonably draw a star with that many points anyway is enough for me. 2*3*5*7*11*13*17*19 ~9.7 million, no number below 20 except for 1,2,3,4,6 has the property that we are looking for. So you can definitively draw any star with up to 9.7 million points except for those with 1,2,3,4,6 points.

Najda

United States3765 Posts

June 13 2014 06:14 GMT

#2111

Okay I get what you're saying now. When you said go 3 further I was actually going 4 so it didn't work. You seem to have solved it though. Worth noting, there should always be an even number of x's that exist, and a pair is what creates a new star. Before I finished reading through your solution, I was thinking about it from a modular division standpoint. n*1 mod j, n*2 mod j, up to n*j mod j should each have a unique solution.

Thanks for the help, maybe I'll be able to sleep now haha.

Thieving Magpie

United States6752 Posts

June 13 2014 06:42 GMT

#2112

On June 13 2014 13:55 Najda wrote:

Show nested quote +

but it doesn't meet the angle requirement.

Even with those parameters its easy to draw a 6 sided star that ends point to point with each point having the same angles as each other.

Simberto

Germany11800 Posts

June 13 2014 06:53 GMT

#2113

Except you now have a superfluous line through the middle of your star breaking symmetry. I guess we really need to have clear definitions what is needed here.

I like the set of n degrees rotational symmetry, straight lines point to point, (not a polygon?) best, as i think that describes a star pretty well.

Mataza

Germany5364 Posts

June 13 2014 06:54 GMT

#2114

On June 13 2014 15:42 Thieving Magpie wrote:

Show nested quote +

Even with those parameters its easy to draw a 6 sided star that ends point to point with each point having the same angles as each other.

Isn't this basically cheating(besides being ugly, how about you copy/paste the star and then add a new line to the copy)?
You added 2 inner lines that do not appear as edges of the star and you lack the third inner line to make those rotational symmteric.
I do know that achieving the third line is impossible without breaking any (unspoken) rule, like not drawing one line twice.