Seriously! Think Outside The Box!
What is the maximum number of equilateral triangles you can create using 10 sticks of equal length without altering the sticks, nor overlapping them?
The answer is 10.
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2 solutions
Would people in "logic" know about simplexes in higher dimensions? Maybe this should be in "geometry"?
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Let alone the interesting philosophical point as to whether being able to imagine a 4-simplex is sufficient to create one. Is existence as mathematical truth a sufficient act of creation? How much of a Platonist do you feel today?
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Well, Mark, I did have the advantage of having already examined the matter of simplexes before on Brilliant. See A Geometric Higher Dimensional Phenomenon? First, I imagined 9 sticks in 3D to get 7 triangles, and when I found out that wasn't enough, going to a 5-Simplex was the next thing. I was kind of surprised to find out that Sharky Kesa was already familiar with the subject.
My answer in that other posted problem explains why there's no easy way to imagine a 5-Simplex as being the vertices of some hypercube, and that's probably why it's not easy to imagine a 5-Simplex in 4D space in the first place.
By "Platonist", don't you mean, "one that believes that there are abstract objects that are ideal, timeless forms that are independent of practical reality"? Wouldn't a 5-simplex qualify?
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@Michael Mendrin – Indeed, a 5-simplex would qualify, but so does a 4-simplex. Four spatial dimensions are not part of humdrum existence, after all! To create even a 4-simplex requires a Platonist view of the independent existence of ideal forms. Of course, I am playing Devil's Advocate, here; I am a Platonist in these matters myself. How else could I have answered this question?
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@Mark Hennings – Let's see....yeah, you did answer this question. Fellow Platonist!
@Mark Hennings – Letting aside the name for a while "Platonism" isn't quite necessary for making a valid theoretical predication of higher spatial dimensions than what we perceive. The validity considered in the concept of a spatial 4 dimensional solid has to do with thinking of higher dimensions by analogy with the way things would be perceived by someone who would see lower dimensions than what we see and therefore such an analogy it's the only necessary thing for attesting or speaking of a thinking construct such as greater spatial dimensions independent of whatever philosophical view we have of them , of either they are or not "existent". Therefore we're free to hypothese and sometimes even indicated so in treating problems of great abstraction that require such constructs of greater dimension the only criteria here being by our hypothesis the validity of logical structure of predication. Therefore if done consistently preserving the rules of correct inference the analogy at hand can then be extended to any number of dimensions if the thinking require for such extensions of the umberof dimensions proves to be based on rules which shall apply for any increased dimension. Nonetheless observe that doing so we are being forced to eliminate from the concept of dimension any intuit content regarding it and claim our understanding of higher dimensions just in terms of the formal and manifest characteristics higher dimensions would imply. Therefore so to say to abstract and formalize in lack of intuition of thought. As such the problem is consistent and valid for Platonists and non-platonists alike because the single difference between them is attesting the ideal reality which a non platonsit might very well deny saying we are speaking of dimensions as extended concepts which alck any reality apart from the formal perception of our mind which is the only responsible thing for conceiving such a concept and they would be right. Moreover considering this view it can be easily showed the deterministic structure underlying the notion of dimension for providing how it can be developed idnependently of other than mechanical computation and formal thinking. And moreover I bet the non-platonist can understand the concept too without making them platonists.
But note though that using the term "platonist" isn't the best term here anyway. For some reason when people say platonism in mathematics they rather refer to idealism which nonetheless is not just the view of Plato nor was first proposed by him and moreover includes some other possible interpretations than the matter of the Forms in the way he said it. Letting transcedental idealism alone any other form of a more esoteric idealism can apply here as for the purpose of the view described by you that mind and thinking are one and have reality problem the only characteristic which matters or imposes relevant is the "ideality" itself so to say. More note that Plato's view doesn't have to be interpreted just as the sort of idealism which says the ideas are really the same or the true deep reality but also in a more realistic form as concepts which are formed in our mind the mind being different from the reality of material world. This problem is rather enormously difficult if possible to solve at all. So the very cute concept of higher than the conceived dimensions doesn't make the things not intelligible just because so to say there's no direct intuition since the inteligible part depends not on the said intuition but on the formal traits considered when speaking of dimensions which make the very perception of the formal predication of a dimension making it a valid concept for both parts idealists or not anyway.
It's difficult for me thinking about 3D, and I just can imagine 4D when the four dimension is the time. I think there is always much platonism everywhere... I get at least 15 equilateral triangles in 3D with 10 sticks of equal length without altering the sticks, nor overlapping them,however we have to also consider equilateral triangles how 3 equidistant points, with "ending vectors" forming angles of 60 degrees. It's left as an exercise to the reader...
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@Guillermo Templado – I have to ask---how do you form 15 equilateral triangles using 10 sticks in 3D? I'm still thinking about that one.
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@Michael Mendrin – There is aproblem. I can't upload a picture like a comment, but I have the picture in my laptop. I don't know how I can upload it. In a next solution of my problems I'll upload it and I'll mention you, for you can see it if you want. They are basically 3 tetrahedrons about a imaginary plane.
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@Guillermo Templado – Something as interesting as that, don't give anything away. Post it. Right now, I can't even wrap my mind around how 3 tetrahedrons can have only and exactly 10 sticks in them---and yet there's 15 faces? I'll wait for the posted problem.
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@Michael Mendrin – Ok, I'm going to do one thing, I'm going to post a solution here... and you decide...
@Guillermo Templado – If you start writing a solution to another problem, and upload a picture to that, you can then cut and paste the markup information into a comment here. You can then abort your solution to that other problem!
Here's one simple way to visualize the 5 vertices of this 5-Simplex. The coordinates of the vertices are
(
1
,
1
,
1
,
0
)
(
1
,
−
1
,
−
1
,
0
)
(
−
1
,
1
,
−
1
,
0
)
(
−
1
,
−
1
,
1
,
0
)
(
0
,
0
,
0
,
5
)
Then the Pythagorean distance between any two vertices is 8
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The regular simplex that I use when I really need coordinates is
(
1
,
0
,
0
,
0
)
,
(
0
,
1
,
0
,
0
)
,
(
0
,
0
,
1
,
0
)
,
(
0
,
0
,
0
,
1
)
along with the point
(
a
,
a
,
a
,
a
)
where
3
a
2
+
(
1
−
a
)
2
=
2
. This generalizes easily to
n
dimensions.
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Yeah, I can see how that works too. The formula for the Pythagorean distance from the outlier vertex to the other 4 are all the same, leading to the quadratic you have for determining a . However, I don't think I would use the word "easily" about generalizing this to n dimensions. I gotta think about that one a bit, and why it should always work. Gimmie a few....
(I'm busy trying to post another problem just now)
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@Michael Mendrin – e 1 , e 2 , ..., e n and a ( e 1 + e 2 + ⋯ + e n ) where ( n − 1 ) a 2 + ( a − 1 ) 2 = 2 o r n a 2 − 2 a − 1 = 0 gives you n + 1 points in R n , any two of which are 2 apart from each other.
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@Mark Hennings – Thanks Mark, I just now got through posting my newest problem. That is interesting, what you and Calvin have just pointed out.
@Mark Hennings – So, basically, if you cut off a corner of a hypercube, you already have a simplex. Hmm...that's interesting. For example, if you cut a corner off a 3D cube, you get 3-simplex, or an equilateral triangle face. Cutting off a corner of a 4D tesseract gets a 4-simplex, or a tetrahedron "face", otherwise known as a "vertex figure", which is always a n-simplex in the case of n-cubes. To form an n+1 simplex, create one extra, outlier point in the fashion Calvin has suggested.
@Guillermo Templado , may I ask why did you delete your solution? I was planning to read Mark's comment in the weekends.
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Please, see Mark Hennings's solution. My solution wasn't right, I had some troubles... not with Mark...
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But the discussion below it was quite interesting!
I agree with Sharky , you deleted the entire discussion.
Not to say you didn't reply to my questions and nonetheless if you were not interested of such matters though you could just have said but the points I pointed were truly or pretty important. Well I don't understand why you didn't took what I wrote seriously. In case you try to mask incoherence in your arguments it's ok with me. I will not reply to your comments as you are not interested in truth. It's a waste of energy , i'll just so to say keep my thoughts for me and speak of them with interested persons anyway.
Rivin's paper shows, amongst much else, that the most triangles that can be achieved in a graph with ( 2 n ) edges is ( 3 n ) , and this occurs when the graph is the complete graph K n on n vertices. Since K 5 can be implemented in 4-dimensional space with all 1 0 triangles equilateral (see Michael Mendrin's comment) the answer to the question is 1 0 .
Ok, reason for my title: We will think dimensions outside the box!
We create a 4D version of a triangular pyramid, (known as a pentachoron). A pentachoron has 10 equal edges and 10 triangular faces.
Now, how do we prove this has the highest number of triangles for the number of edges?
We simply show that in 5D, the shape with the least number of edges has 15 edges (try to prove this), and in 2 and 3D (mixing them up as well), we cannot create 10 or more (brute force method, left as an exercise).