## Four pointed Triangles, the Case for an Irrational Triangle

### Four pointed Triangles, the Case for an Irrational Triangle

Very short and simple argument.

1. A trapezoid is presented.

2. A trapezoid, with a proportion equal in such a manner where the bottom portion is equal to the top of the original, is placed on top.

3. This process repeats indefinitely.

4. A triangle appears to form. The top of the top trapezoid appears as a simple point compared to the base of the bottom trapezoid.

5. It appears as a triangle when in reality it is a trapezoid causing a contradictory 4 pointed triangle to appear.

6. The triangle can thus be argued as a simultaneous trapezoid considering the top point is indefinite.

7. This apex point as indefinite is further proven where a line is halved by a dot. Each new line (within the larger line) in turn is halved...so on and so forth. This halving continual occurs. Each new line is thus equal to a dot of the original line until magnified. Each dot, thus contains a line and each line exist through a dot.

The dot is only observed as a small line upon further magnification, with this line as composed of further "dots" being composed of further lines.

8. Each dot is thus a potential line. So in one respect the triangle can be argued as a potential 4 side figure, 5 side figure or six sided figure.

9. This applies to all geometric figures, where each dot, as composed of further lines which are composed of further lines, necessitates the possibly that each corner or a geometric solid is a geometric ripple of smaller and smaller irrational angles that manifest continually.

1. A trapezoid is presented.

2. A trapezoid, with a proportion equal in such a manner where the bottom portion is equal to the top of the original, is placed on top.

3. This process repeats indefinitely.

4. A triangle appears to form. The top of the top trapezoid appears as a simple point compared to the base of the bottom trapezoid.

5. It appears as a triangle when in reality it is a trapezoid causing a contradictory 4 pointed triangle to appear.

6. The triangle can thus be argued as a simultaneous trapezoid considering the top point is indefinite.

7. This apex point as indefinite is further proven where a line is halved by a dot. Each new line (within the larger line) in turn is halved...so on and so forth. This halving continual occurs. Each new line is thus equal to a dot of the original line until magnified. Each dot, thus contains a line and each line exist through a dot.

The dot is only observed as a small line upon further magnification, with this line as composed of further "dots" being composed of further lines.

8. Each dot is thus a potential line. So in one respect the triangle can be argued as a potential 4 side figure, 5 side figure or six sided figure.

9. This applies to all geometric figures, where each dot, as composed of further lines which are composed of further lines, necessitates the possibly that each corner or a geometric solid is a geometric ripple of smaller and smaller irrational angles that manifest continually.

### Re: Four pointed Triangles, the Case for an Irrational Triangle

I understood that! Eod this is a special day indeed.

You are stacking an endless sequence of similar trapezoids on top of each other, each one of smaller size than the one below.

If the size (measured any way you like) happens to go to zero, then "at infinity" you'd conceptually have a point; that is, the sequence of trapezoids converges to a triangle with the top point at infinity.

I don't know anything about projective geometry, but that's a branch of math exactly designed to reason about this kind of situation. There is a "point at infinity" where parallel lines converge, and you can do geometry in that kind of space. Projective geometry is actually based on the theory of perspective as developed during the Renaissance.

https://en.wikipedia.org/wiki/Projective_geometry

I also wanted to mention that not every such tower of trapezoids converges to a point at the top. For example suppose the dimension of the top of the first trapezoid (call it trapezoid 1) is 1, and the next one up (trapezoid 2) is 3/4, and the next one (trapezoid 3) is 5/8, and the next 11/16, and in general the top of trapezoid n is (2n - 1)/2^n, then the limiting trapezoid has top edge 1/2. In other words it's a decreasing sequence that doesn't happen to converge to zero. That can happen.

I think it's pretty cool that I understood something you said and I hope you found my response helpful.

You are stacking an endless sequence of similar trapezoids on top of each other, each one of smaller size than the one below.

If the size (measured any way you like) happens to go to zero, then "at infinity" you'd conceptually have a point; that is, the sequence of trapezoids converges to a triangle with the top point at infinity.

I don't know anything about projective geometry, but that's a branch of math exactly designed to reason about this kind of situation. There is a "point at infinity" where parallel lines converge, and you can do geometry in that kind of space. Projective geometry is actually based on the theory of perspective as developed during the Renaissance.

https://en.wikipedia.org/wiki/Projective_geometry

I also wanted to mention that not every such tower of trapezoids converges to a point at the top. For example suppose the dimension of the top of the first trapezoid (call it trapezoid 1) is 1, and the next one up (trapezoid 2) is 3/4, and the next one (trapezoid 3) is 5/8, and the next 11/16, and in general the top of trapezoid n is (2n - 1)/2^n, then the limiting trapezoid has top edge 1/2. In other words it's a decreasing sequence that doesn't happen to converge to zero. That can happen.

I think it's pretty cool that I understood something you said and I hope you found my response helpful.

### Re: Four pointed Triangles, the Case for an Irrational Triangle

I really don't know what to say quite frankly...wtf.wtf wrote: ↑Wed Oct 16, 2019 12:28 amI understood that! Eod this is a special day indeed.

You are stacking an endless sequence of similar trapezoids on top of each other, each one of smaller size than the one below.

If the size (measured any way you like) happens to go to zero, then "at infinity" you'd conceptually have a point; that is, the sequence of trapezoids converges to a triangle with the top point at infinity.

I don't know anything about projective geometry, but that's a branch of math exactly designed to reason about this kind of situation. There is a "point at infinity" where parallel lines converge, and you can do geometry in that kind of space. Projective geometry is actually based on the theory of perspective as developed during the Renaissance.

https://en.wikipedia.org/wiki/Projective_geometry

I also wanted to mention that not every such tower of trapezoids converges to a point at the top. For example suppose the dimension of the top of the first trapezoid (call it trapezoid 1) is 1, and the next one up (trapezoid 2) is 3/4, and the next one (trapezoid 3) is 5/8, and the next 11/16, and in general the top of trapezoid n is (2n - 1)/2^n, then the limiting trapezoid has top edge 1/2. In other words it's a decreasing sequence that doesn't happen to converge to zero. That can happen.

I think it's pretty cool that I understood something you said and I hope you found my response helpful.

Anyhow as to your second paragraph from the bottom.

Yeah, they don't converge to a point...that is the point. That is the paradox as well. They never converge, as a matter of fact they just keep replicating ad-infinitum.

But we are still left with the trapezoids existing within a point. If I can see only x number of trapezoids, before they converge to a point, but decided to "scroll up", any set of x number of trapezoids will always look the same as the first original set.

A 4 point triangle is dynamic and observes that considering each "set" of x trapezoids is proportionally the same as the original set, when both are viewed on there own terms, the triangle can represent:

Each fractal set of shapes, existing as the same proportions to eachother as the beginning set, fundamentally are a "further distance version" of the original set.

Each fractal as a standalone can be observed as the first set superpositioned at a distance.

### Re: Four pointed Triangles, the Case for an Irrational Triangle

I'm not entirely sure what's significant about this in your mind.

You can think about the situation in reverse. Start with any isosceles triangle.

Any horizontal line between the vertex and the base defines an isosceles trapezoid; and if we imagine an infinite sequence of horizontal lines, each one say half way between the base of its trapezoid and the vertex, we do indeed get a stack of similar isosceles trapezoids whose union is indeed the original isosceles triangle.

Is that an example of what you're thinking of? What is the significance of this for you? There is no "trapezoid in the point" as you put it. There's just an isosceles triangle partitioned into a stack of isosceles trapezoids.

Here's a little picture I drew. To make everything nice and simple consider an equilateral triangle of height 1. A horizontal (dotted) line at height 1/2 makes an isosceles trapezoid. Another horizontal line 1/4 above that makes another one, and so forth. The base angles are all equal since the horizontals are parallel, so all of these trapezoids are similar (in proportion as you put it).

It's perfectly true that we can thereby partition the original triangle into infinitely many similar trapezoids. It's a bit of a curiosity perhaps but there is nothing mysterious going on. There are no trapezoids inside the point. It's just that there is an infinite sequence of trapezoids. The top of the trapezoids converge to the apex of the triangle. It's true, and fun to contemplate, but not particularly meaningful IMO.

The point at the top is indeed the limit of the line segments that make up the tops of the trapezoids. But that doesn't make the point a little line segment. On the contrary, it's a point. This is no different than the line segments on the real line [-1, 1], [-1/2, 1/2], [-1/4 1/4], ... converging to the single point 0. Happens all the time. Not mysterious. The point of convergence is not a little line segment. It's a zero dimensional point.

Image hosting courtesy https://postimages.org/.

You can think about the situation in reverse. Start with any isosceles triangle.

Any horizontal line between the vertex and the base defines an isosceles trapezoid; and if we imagine an infinite sequence of horizontal lines, each one say half way between the base of its trapezoid and the vertex, we do indeed get a stack of similar isosceles trapezoids whose union is indeed the original isosceles triangle.

Is that an example of what you're thinking of? What is the significance of this for you? There is no "trapezoid in the point" as you put it. There's just an isosceles triangle partitioned into a stack of isosceles trapezoids.

Here's a little picture I drew. To make everything nice and simple consider an equilateral triangle of height 1. A horizontal (dotted) line at height 1/2 makes an isosceles trapezoid. Another horizontal line 1/4 above that makes another one, and so forth. The base angles are all equal since the horizontals are parallel, so all of these trapezoids are similar (in proportion as you put it).

It's perfectly true that we can thereby partition the original triangle into infinitely many similar trapezoids. It's a bit of a curiosity perhaps but there is nothing mysterious going on. There are no trapezoids inside the point. It's just that there is an infinite sequence of trapezoids. The top of the trapezoids converge to the apex of the triangle. It's true, and fun to contemplate, but not particularly meaningful IMO.

The point at the top is indeed the limit of the line segments that make up the tops of the trapezoids. But that doesn't make the point a little line segment. On the contrary, it's a point. This is no different than the line segments on the real line [-1, 1], [-1/2, 1/2], [-1/4 1/4], ... converging to the single point 0. Happens all the time. Not mysterious. The point of convergence is not a little line segment. It's a zero dimensional point.

Image hosting courtesy https://postimages.org/.

### Re: Four pointed Triangles, the Case for an Irrational Triangle

wtf wrote: ↑Wed Oct 16, 2019 3:58 amI'm not entirely sure what's significant about this in your mind.

Alot is significant:

1. Points contain forms, and are not strictly "empty". If we understand point space, at the subjective and objective level we understand ourselves and environment around us. "Getting to the point of life" may actually be more than just a saying but an expression of the subconscious.

2. It necessitates distance is strictly fractals, an object can be in two or more positions at the same time where the point acts as a quantum medium. At the same time it also observes how an identity, empirical or subjectively abstract, is stretched through time much like a band.

Again point 1.

It gives us a better understanding of how to approach time at both the subjective level and objective level. It is, pardon the pun, a "point" to meditate on when understanding the nature of reality and how to approach it. This geometric "proof" is just a proof, but also an analogy to other phenomenon.

Trapezoids have, according to some pseudo scientific theories, an effect on the psyche because of its irrational angles. It induces madness in some, and creativity in others....but the line is blurred between the two. The "theory" is because it irrational it causes psychic fracturing...but again these are unproven theories.

3. Fractals (fractions by default) observe that any projection of a phenomenon across time is divisive by nature. One identity results in a fraction of itself along a timeline, dually this new identity while entropied, is still proportional to the original. Size is grounded in time.

You can think about the situation in reverse. Start with any isosceles triangle.

lol, I have and still am.

Any horizontal line between the vertex and the base defines an isosceles trapezoid; and if we imagine an infinite sequence of horizontal lines, each one say half way between the base of its trapezoid and the vertex, we do indeed get a stack of similar isosceles trapezoids whose union is indeed the original isosceles triangle.

Is that an example of what you're thinking of? What is the significance of this for you? There is no "trapezoid in the point" as you put it. There's just an isosceles triangle partitioned into a stack of isosceles trapezoids.

Here's a little picture I drew. To make everything nice and simple consider an equilateral triangle of height 1. A horizontal (dotted) line at height 1/2 makes an isosceles trapezoid. Another horizontal line 1/4 above that makes another one, and so forth. The base angles are all equal since the horizontals are parallel, so all of these trapezoids are similar (in proportion as you put it).

It's perfectly true that we can thereby partition the original triangle into infinitely many similar trapezoids. It's a bit of a curiosity perhaps but there is nothing mysterious going on. There are no trapezoids inside the point. It's just that there is an infinite sequence of trapezoids. The top of the trapezoids converge to the apex of the triangle. It's true, and fun to contemplate, but not particularly meaningful IMO.

I understand why you can believe the dot is not meaningful from a traditional stance. However I am not much one for tradition.

So...

Again we are left with a paradox...a real simple one:

If you keep "scrolling up" the trapezoid, the trapezoid unfolds from the point and folds back to nothing through the base points.

1. This shows recursion of forms is a wave function: form-no form-form-no form, etc.

2. The point acts much like a field and becomes polarized (divided into localized points) through forms. The forms are literally just made up.

3. The point is potential form.

4. If the trapezoids continually progress, that necessitates (using your drawing which is the picture I had in mind) each line shrinking an expanding as it is "scrolled down". The infinite progression of trapezoids are looping themselves through the beginning and end points....so something as simple as walking forward is pulling yourself through space by a self loop...the phenomenon is acting link a wheel so to speak...thus technically an object may not be moving anywhere but space is moving around it.

So you have a man walking across the street. But it is also possible to interpret it as the man is pulling the street and surrounding phenomenon through himself....see the difference and how it is rooted in geometry (considering phenomenon are 99.9999 space)? Do you see where I am going?

The bottom trapezoid effectively disappears, as the progression occurs, but returns in a fractal form "up top" until it goes back to it's original position. So using your drawing, which I appreciate because it makes things simpler to explain, this "triangle" is a set of trapezoids (we will say 7 for examples sake) that continually repeat.

Progression is a form of being pulled away from itself and eventually loops back....each form is literally spinning itself through space because of its repition.

5. Each phenomenon, as repeating, is thus a wave superpositioned on top of another wave.

Quantum waves equate to a "primordial ocean" that "being" floats on....metaphorically speaking, but even these metaphors as forms of the unconscious...another primordial ocean.

6. This sets a premise that in theory you can have 0 point energy....emphasis on the word "theory".

However it sets up equal questions about the nature of the unconscious and empty slate mind as well as the nature of "darkness".

The point at the top is indeed the limit of the line segments that make up the tops of the trapezoids. But that doesn't make the point a little line segment. On the contrary, it's a point. This is no different than the line segments on the real line [-1, 1], [-1/2, 1/2], [-1/4 1/4], ... converging to the single point 0. Happens all the time. Not mysterious. The point of convergence is not a little line segment. It's a zero dimensional point.

The point is a localized position where forms expand and contract from. When you observe a point you are observing the convergence and divergence in forms...it is synthetic in nature.

Image hosting courtesy https://postimages.org/.

### Re: Four pointed Triangles, the Case for an Irrational Triangle

wtf wrote: ↑Wed Oct 16, 2019 3:58 amI'm not entirely sure what's significant about this in your mind.

Alot is significant:

1. Points contain forms, and are not strictly "empty". If we understand point space, at the subjective and objective level we understand ourselves and environment around us. "Getting to the point of life" may actually be more than just a saying but an expression of the subconscious.

2. It necessitates distance is strictly fractals, an object can be in two or more positions at the same time where the point acts as a quantum medium. At the same time it also observes how an identity, empirical or subjectively abstract, is stretched through time much like a band.

Again point 1.

It gives us a better understanding of how to approach time at both the subjective level and objective level. It is, pardon the pun, a "point" to meditate on when understanding the nature of reality and how to approach it. This geometric "proof" is just a proof, but also an analogy to other phenomenon.

Trapezoids have, according to some pseudo scientific theories, an effect on the psyche because of its irrational angles. It induces madness in some, and creativity in others....but the line is blurred between the two. The "theory" is because it irrational it causes psychic fracturing...but again these are unproven theories.

3. Fractals (fractions by default) observe that any projection of a phenomenon across time is divisive by nature. One identity results in a fraction of itself along a timeline, dually this new identity while entropied, is still proportional to the original. Size is grounded in time.

You can think about the situation in reverse. Start with any isosceles triangle.

lol, I have and still am.

Any horizontal line between the vertex and the base defines an isosceles trapezoid; and if we imagine an infinite sequence of horizontal lines, each one say half way between the base of its trapezoid and the vertex, we do indeed get a stack of similar isosceles trapezoids whose union is indeed the original isosceles triangle.

Is that an example of what you're thinking of? What is the significance of this for you? There is no "trapezoid in the point" as you put it. There's just an isosceles triangle partitioned into a stack of isosceles trapezoids.

Here's a little picture I drew. To make everything nice and simple consider an equilateral triangle of height 1. A horizontal (dotted) line at height 1/2 makes an isosceles trapezoid. Another horizontal line 1/4 above that makes another one, and so forth. The base angles are all equal since the horizontals are parallel, so all of these trapezoids are similar (in proportion as you put it).

It's perfectly true that we can thereby partition the original triangle into infinitely many similar trapezoids. It's a bit of a curiosity perhaps but there is nothing mysterious going on. There are no trapezoids inside the point. It's just that there is an infinite sequence of trapezoids. The top of the trapezoids converge to the apex of the triangle. It's true, and fun to contemplate, but not particularly meaningful IMO.

I understand why you can believe the dot is not meaningful from a traditional stance. However I am not much one for tradition.

So...

Again we are left with a paradox...a real simple one:

If you keep "scrolling up" the trapezoid, the trapezoid unfolds from the point and folds back to nothing through the base points.

1. This shows recursion of forms is a wave function: form-no form-form-no form, etc.

2. The point acts much like a field and becomes polarized (divided into localized points) through forms. The forms are literally just made up.

3. The point is potential form.

4. If the trapezoids continually progress, that necessitates (using your drawing which is the picture I had in mind) each line shrinking an expanding as it is "scrolled down". The infinite progression of trapezoids are looping themselves through the beginning and end points....so something as simple as walking forward is pulling yourself through space by a self loop...the phenomenon is acting link a wheel so to speak...thus technically an object may not be moving anywhere but space is moving around it.

So you have a man walking across the street. But it is also possible to interpret it as the man is pulling the street and surrounding phenomenon through himself....see the difference and how it is rooted in geometry (considering phenomenon are 99.9999 space)? Do you see where I am going?

The bottom trapezoid effectively disappears, as the progression occurs, but returns in a fractal form "up top" until it goes back to it's original position. So using your drawing, which I appreciate because it makes things simpler to explain, this "triangle" is a set of trapezoids (we will say 7 for examples sake) that continually repeat.

Progression is a form of being pulled away from itself and eventually loops back....each form is literally spinning itself through space because of its repition.

5. Each phenomenon, as repeating, is thus a wave superpositioned on top of another wave.

Quantum waves equate to a "primordial ocean" that "being" floats on....metaphorically speaking, but even these metaphors as forms of the unconscious...another primordial ocean.

6. This sets a premise that in theory you can have 0 point energy....emphasis on the word "theory".

However it sets up equal questions about the nature of the unconscious and empty slate mind as well as the nature of "darkness".

The point at the top is indeed the limit of the line segments that make up the tops of the trapezoids. But that doesn't make the point a little line segment. On the contrary, it's a point. This is no different than the line segments on the real line [-1, 1], [-1/2, 1/2], [-1/4 1/4], ... converging to the single point 0. Happens all the time. Not mysterious. The point of convergence is not a little line segment. It's a zero dimensional point.

The point is a localized position where forms expand and contract from. When you observe a point you are observing the convergence and divergence in forms...it is synthetic in nature. Void is creative and destructive.

If I take that picture you drawn, and take the end points and move them up and down, the picture not only expands but also necessitates the 7 trapezoid existing as...we will say 10 or 12 for examples sake....or the new set of trapezoids changes Sizes compared to the original set.

Thus each points position determines the position of the phenomenon as well, but not just is position but form. Each set of localized points determines how reality unfolds from a backdrop of a point space field (quantum waves).

If you applied this geometry to an empirical form (or even in the psyche as a meditative exercise) the form effectively pulls itself through reality.

It would be like watching rain fall on top of a roof. It collects in streams due to the angulature and those streams converge at a point (the corner angle) and diverge again from that same point. So is the rain falling on the roof...or is the roof pulling itself through a series of chaotic fragmented movements. If you fixed the rain as still, the roof is moving through it. If you fix the roof as still, the rain is being redirected.

Either way, the angulature redirects the movements causing them to expand and contract. Geometric forms bend the surround space.

Image hosting courtesy https://postimages.org/.

Points are dynamic.

### Re: Four pointed Triangles, the Case for an Irrational Triangle

"I understand why you can believe the dot is not meaningful from a traditional stance. However I am not much one for tradition."

It's not a dot, it's a mathematical point. It has no internal structure.

If you feel otherwise that's your right, but that's not in accordance with usual math. And I'm afraid I can't follow the logic of your alternative math.

It's not a dot, it's a mathematical point. It has no internal structure.

If you feel otherwise that's your right, but that's not in accordance with usual math. And I'm afraid I can't follow the logic of your alternative math.

### Re: Four pointed Triangles, the Case for an Irrational Triangle

And tell me what the difference between a dot "•" and a mathematical point "•".wtf wrote: ↑Thu Oct 17, 2019 4:26 am"I understand why you can believe the dot is not meaningful from a traditional stance. However I am not much one for tradition."

It's not a dot, it's a mathematical point. It has no internal structure.

If you feel otherwise that's your right, but that's not in accordance with usual math. And I'm afraid I can't follow the logic of your alternative math.

Actually it doesn't have to have an internal structure, it just has to act as a quantum medium to another phenomenon. As such it is a door way and as a door way "contains" other phenomenon.

The continual progression of the trapezoids necessitates them unfolding from the top 0d point and folding back into the bottom. Considering the word "folding" will confuse you, expansion and contraction will work.

You can't follow the logic of your own math. "Infinite 0d points make a line"? But this results in either 0 having a length, which it doesn't as the unit interval is a length, or infinite 0d points as having infinite lines.

The point as formless is 0d. The point as form is 1d (line). Nothing is formlessness. Being is form.

You have no grounding.

### Re: Four pointed Triangles, the Case for an Irrational Triangle

Actually not really.

According to the dictionary (simple Google the word):

Dot is defined as a spot

A spot is defined as a point or position.

Both the dot and point are positions. Considering the 0d point is graphed, in a unit interval, you end up contradicting yourself. The unit interval is defined by dots not 0d points.

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