How many equilateral tetrahedra can be connected together without any empty space between them?

[Olga]

How many points are needed to create the smallest equilateral tetrahedron?

 

[exchemist]

Tetrahedron packing - Wikipedia

en.wikipedia.org

 

[James R]

How many points are needed to create the smallest equilateral tetrahedron?

All tetrahedrons have four vertices, by definition.

 

[Olga]

All tetrahedrons have four vertices, by definition.

Это будет тетраэдр?

 

[James R]

Will it be a tetrahedron?

An equilateral tetrahedron has four faces with equal areas, by definition. Each face shares one side with each of the other three faces.

It sounds like you might be confused about what a tetrahedron is, Olga. Do you know?

 

[Olga]

An equilateral tetrahedron has four faces with equal areas, by definition. Each face shares one side with each of the other three faces.

It sounds like you might be confused about what a tetrahedron is, Olga. Do you know?

Не смеши меня, Джеймс. Я знаю, что такое тетраэдр. Итак: мы имеем пирамиду с основанием состоящим из 3-х точек. Над ними вершина пирамиды, состоящая из 1-й точки. Опустим отрезок на основание. Куда он попадёт? Между точками? Значит, в основании пирамиды помещается ещё одна точка, и она уже не состоит из 4-х точек, а минимум из 5.

 

[James R]

I know what a tetrahedron is.

Then I don't understand why you are so confused about how many faces and vertices a tetrahedron has.

So: we have a pyramid with a base consisting of 3 points. Above them is the top of the pyramid, consisting of the 1st point. Let's lower the segment to the base.

If you do that, it won't be an equilateral tetrahedron any more.

Do you understand what "equilateral" means?

Do you want to talk now about irregular tetrahedrons? If that's what you wanted to talk about, why didn't you say so?

Where will it go? Between the points?

It will go wherever you want it to go, I suppose. Somewhere on the base, like you said.

But then, with all 4 vertices in a single plane, you no longer have a tetrahedron. Instead, you have a flat shape made of four triangles.

So, at the base of the pyramid there is another point, and it no longer consists of 4 points, but at least 5.

Where did point number 5 come from?

You're not being very clear, Olga.

 

[DaveC426913]

So: we have a pyramid with a base consisting of three points.

Those three points are vertices of the tetrahedron.

A vertex is defined as a point where two or more edges terminate.

Above them is the apex of the pyramid, consisting of one point.

This the fourth vertex.

Let's drop a line segment onto the base. Where does it end up? Between the points? That means another point fits at the base of the pyramid,

Sure. So?

What do you mean by "another" point? It is already an infinity of other points on that surface.


A triangular polygon (of which a tetrahedron has four) has exactly three edges. Adding - or defining - points on it does not change the number of edges of a triangle. Thus, a tetrahedron is still a tetrahedron.

and it no longer consists of four points, but at least five.

This is your error. It never consisted of merely four points.

Any shape contains an infinite number of points. Every surface is an infinite number of points, regardless of how you position them.




But in a tetrahedron only four of them are vertices.

That is what defines it is a polyhedron.



If, instead, Olga meant to add a fifth vertex, then indeed, this is no longer an equilateral tetrahedron.


This is a different shape.



Perhaps Olga was confused about the difference between a point and a vertex. Now she knows better.

 

[James R]

Perhaps Olga was confused about the difference between a point and a vertex. Now she knows better.

Maybe we'll find out when she gets back from her break.

The question in the opening post doesn't make much sense, as written, and Olga hasn't elaborated on what she meant yet. Maybe she doesn't know what she meant.

 

[Olga]

Возможно, мы узнаем, когда она вернется с отдыха.

Вопрос в первом сообщении в том виде, в котором он написан, не имеет особого смысла, и Ольга пока не объяснила, что имела в виду. Возможно, она сама не понимает, что имела в виду.

Как вы можете доказать, что это именно тетраэдр, если не знаете его настоящий размер? Бесконечное количество точек подразумевает всё что угодно, не так ли?

 

[DaveC426913]

How can you prove it's a tetrahedron if you don't know its true size?

The definition of a tetrahedron does not include size specificatons.
Thus, size is not required to prove it.


Let's take a simpler example:


"How can you prove this is a triangle if you don't know its true size?"

Do you see why the question makes no sense?

An infinite number of points implies anything, doesn't it?

No it does not.

There are an infinite number of points in the equation y=1.
In the xy plane, y=1 is a line.
No matter how many points you define, it still forms a line.

In the xyz volume, y=1 is a plane.
No matter how many points you define, it still forms a plane.

Even less ambitious:
In the xy plane, y=1 where 0<x<1, is a line segment.
No matter how many points you define, it still forms a line segment of length 1.

Do they not teach basic geometry in Russia? How did you get to post secondary education without basic geometry?

 

[Olga]

The definition of a tetrahedron does not include size specificatons.
Thus, size is not required to prove it.


No it does not.

There are an infinite number of points in the equation y=1.
In the xy plane, y=1 is a line. No matter how many points you define, it still forms a line.

In the xyz volume, y=1 is a plane. No matter how many points you define, it still forms a plane.

Do they not teach basic geometry in Russia? How did you get to post secondary education without basic geometry?

Ха, я геометрию экстерном сдала на отлично, за один день весь учебник прочла, как роман. Геометрия - один из моих любимых предметов.
Как это "нет размеров", Дэйв? Тогда как вы отличаете тетраэдр от куба, например?

 

[DaveC426913]

Ha, I passed geometry as an external student with flying colors, and I read the entire textbook in one day, like a novel.

You should have read it slower. Clearly, you did not learn it because you are asking questions that would have been answered by your book.

Geometry is one of my favorite subjects.
What do you mean "no dimensions," Dave?

This is a translation error. At no time did I say "no dimensions".

Then how do you distinguish a tetrahedron from a cube, for example?

Let's start with a simpler example:

Here is a triangle:


It contains an infinite number of points.

Are you unable to distinguish it from a square?
Do you need to know its "true size" to conclude that it is a triangle and not a square?

 

[Olga]

You should have read it slower. Clearly, you did not learn it because you are asking questions that would have been answered by your book.


This is a translation error. At no time did I say "no dimensions".


Let's start with a simpler example:

Here is a triangle:
View attachment 7203

It contains an infinite number of points.

Are you unable to distinguish it from a square?
Do you need to know its "true size" to conclude that it is a triangle and not a square?

Дэйв, изначальный вопрос имел примерно такой смысл: при каких размерах обьекты перестают иметь форму? Вам лучше было бы ответить что то типа "при планковских размерах". Но тогда я бы спросила: а при планковских размерах имеют ли объекты какую-либо форму?

 

[DaveC426913]

Dave, the original question was something like this: at what size do objects cease to have shape? A better answer would be something like "at Planck sizes." But then I'd ask: do objects have any shape at Planck sizes?

This is a category error.

At one point you are talking about mathematical objects, and at another you are talking about physical forms which are made of atoms and thus have a lower bound on shape.

In mathematics, it is trivial to have a tetrahedron whose side length is infinitesimally short. Yes, it is still a tetrahedron.

If you have ever examined the Mandelbrot Set (a mathematical object) you know you can zoom into it essentially forever, until you are looking at a section of it that must be far, far smaller than the Plank length, yet it still has a wealth of detail.

 

[Olga]

This is a category error.

At one point you are talking about mathematical objects, and at another you are talking about physical forms which are made of atoms and thus have a lower bound on shape.

In mathematics, it is trivial to have a tetrahedron whose side length is infinitesimally short. Yes, it is still a tetrahedron.

If you have ever examined the Mandelbrot Set (a mathematical object) you know you can zoom into it essentially forever, until you are looking at a section of it that must be far, far smaller than the Plank length, yet it still has a wealth of detail.

Математика - это инструмент, который использует физика. И да, я имела ввиду реальный, физический тетраэдр.

 

[DaveC426913]

Mathematics is a tool used by physics.

Maybe so but a tetrahedron is defined mathematically, not by the number of atoms a given tetrahedron might contain.

And yes, I meant a real, physical tetrahedron.

Then you have your answer.

 

[Olga]

Maybe so but a tetrahedron is defined mathematically, not by the number of atoms a given tetrahedron might contain.


Then you have your answer.

Не совсем. Если у нашего тетраэдра каждая грань имеет планковскую длину, то чему тогда равно расстояние между вершиной и основанием?

 

[DaveC426913]

Not quite. If each face of our tetrahedron has a Planck length, then what is the distance between the vertex and the base?

That is poorly defined.
A face does not have a clearly defined length unless you define what you mean.

But it is certainly possible to have a tetrahedron that is a Planck length along each side.

 

[Olga]

That is poorly defined.
A face does not have a clearly defined length unless you define what you mean.

But it is certainly possible to have a tetrahedron that is a Planck length along each side.

Т.е. самый маленький тетраэдр будет иметь грани, равные планковской длине каждая?