Triangular Numbers (equilaterals)

[west bronze]

did you review rafain's demonstration?

[fallen trellis]

Omega gave a proof with regards to the sequence

[west bronze]

the question is whether the OP is correct

[fallen trellis]

The sum of the nth term and n+1th term of the sequence is equal to the n+1 th square

[west bronze]

and whether the demonstrations provided in this thread either support or refute the main proposition

[fallen trellis]

He just showed the visual picture

[west bronze]

what do you mean?

[fallen trellis]

You have to take all of this for granted first to proceed like that

[west bronze]

his units from figure to figure were not consistent

and i have no knowledge of what he was even trying to demonstrate

[fallen trellis]

There is a closed form representation of the n th triangular number

[west bronze]

i think he wanted to prove that two consecutive triangular numbers = a square

which is false

as I showed

[fallen trellis]

Nope

Wait let me check what I'm saying

[west bronze]

there is no need to go up into the stratosphere on this

take 3 and 6

two consecutive triangular numbers

they do not form a square

[fallen trellis]

Edit was unnecessary what I said first was finr

[west bronze]

because they are actually derived from two equilateral triangles as demonstrated by Nicomachus

[fallen trellis]

9 is a square

[west bronze]

no my friend

[fallen trellis]

And yes there is

[snow hamlet]

Upon this derivation, are the numbers still bound to the equilateral configurations and their geometric properties?

[fallen trellis]

Abstraction makes more clear

[west bronze]

9 is a square if you are building your square from two* right* triangles

two eright triangles make a square

[fallen trellis]

Your only refutation is if you misunderstand the definitions or you make your own definitions

[west bronze]

right triangles

[fallen trellis]

I thought it was triangular not triangles

[west bronze]

well i know what a triangular number is

[fallen trellis]

They are unrelated to triangles

[west bronze]

it is a number derived from an equilateral shape

it can not produce a square

everyone has stubbornly refused to depart from convention

even when the convention is illogical and unacceptable

the convention of a square number does not hold

it must be abandoned for a moment

the idea of a square number in the context of this discussion has no plae

square numbers do not exist for this situation

[fallen trellis]

Not if you formally define things which you are not

[west bronze]

it has no place here

it has no reality

it cannot be constructed

people tried

and they failed

miserably

[fallen trellis]

Mathematics is a formal system

[west bronze]

yes

[fallen trellis]

It is by definition applied logic

[west bronze]

and triangualrs have a history

and they are a system

[fallen trellis]

If you don't want to work under those constraints you're free to do whatever you want

[west bronze]

and squares do not exist here

you cannot use them for demonstraion here

[fallen trellis]

https://en.m.wikipedia.org/wiki/Principle_of_explosion#:~:text=In symbolic logic%2C the principle,follows that Q is true.

[west bronze]

because they have no relation to triangular numbers

and insisting upon adherence to this convention causes many many misinterpretation of these kinds of numbers.

what can you do with a square number in this present context?

nothing

you are jsut looking at the num bers 46 9 16 25

[fallen trellis]

Set P: the set of square integers
x in P iff there exists a integer y such that y*y=x

[west bronze]

and referring to a convention you are declaring them to be squares when in fact they are NOT

I go to lunch

[glass echo]

Triangular Numbers (Abandon all hope ye who enters here)

[fallen trellis]

You're overinflating the importance of semantic crutches

Read some wittgenstein or just any philosophy of language for that matter

[glass echo]

fixed the title btw

just as a warning

[snow hamlet]

In concise terms, triangular numbers are integers which are expressible in such equilateral configurations
Nothing is said of whether such triangular numbers are expressible in other configurations, so I don't really understand the reluctance to consider e.g. right-angled configurations
The fixation on lengths and areas is also extraneous at best

[west bronze]

ok

you are one of those who produced a poor demonstration against the OP because you did not conceive of these numbers as being equilateral triangles. There are no right angles here. Bringing them in is what is causing problems. They just DON'T exist. You cannot make them exist here. All you can work with is 60 and 120 degrees, We do not have 90 degrees here. That only led all of you into incorrect and false demonstrations.

We are studying geometry and math.

Simply relate these numbers to the shapes we are looking at.

[snow hamlet]

Why?

[west bronze]

there are no square shapes in this math.

[snow hamlet]

We already have the numbers, why are the geometric properties still relevant?

[fallen trellis]

There are no triangles either

Study Euclid properly first

[west bronze]

that is it in a nutshell. You dont accept the shapes these numbers are derived from.

[fallen trellis]

You don't appear to have had a proper grounding in classical geometry first and foremost

[west bronze]

and because you dont accept the sahpes we are not communicating

there are NO right triangles or squares in this math.

it does not work

[snow hamlet]

I accept that they are the shapes from which the numbers are derived, but the shapes have served their purpose once the numbers are derived.

[west bronze]

we want to investigate triangular numbers

[snow hamlet]

We want to investigate triangular numbers

[west bronze]

we are looking at objects that exist in this geometry and math

[snow hamlet]

We are looking at numbers

[west bronze]

it's not just math

look at what Nicomachus is teching

teaching

[snow hamlet]

Yeah look at it

[west bronze]

it is geometry and math

I have read his work about 10 times

I have read euclid 10 times

I gotta go

[snow hamlet]

Geometry serving the purpose of deriving the sequence, and ceasing once that's done

[snow hamlet]

Good

[glass echo]

is he rejoining the server after every time he goes away?

[fallen trellis]

You don't read mathematics

[snow hamlet]

Every time he finds interest in another ill-fated debate

[fallen trellis]

You do it instead

No wonder

[glass echo]

i find it sad that the most discussed topics are rockhoven's

[fallen trellis]

Yea

Prime example of scaring the hoes away

[snow hamlet]

He does not commit any violation to the rules, so the mods/management can't exactly punish him
I will keep to this approach: Brief, to-the-point refutations
@fallen trellis You should consider the yield to such dedication of time

[fallen trellis]

Yea

I just had nothing to do for 30 mins

I don't think the main point even sunk in tho

[glass echo]

He does not commit any violation to the rules
i wish i could just metaphorically plant drugs in his car and arrest him for that

...not a good sign when a moderation manager says that

[snow hamlet]

What desperation does to middle management

[glass echo]

im probably an upper management though, i kneel only before yoavmal and head manager

[snow hamlet]

You work, that sets you apart from upper management

[glass echo]

welp

[snow hamlet]

I just realized, by that definition, yoav alone is upper management
Viata and her workload, sometimes I wonder how med students juggle all that

[glass echo]

its insane

[hexed lake]

Q

[snow hamlet]

I would like this pinned - the irony, oh so delicious!

[west bronze]

https://en.wikipedia.org/wiki/Square#/media/File:Regular_polygon_4_annotated.svg

Here is a square

resolve this square into two equilateral triangles

https://en.wikipedia.org/wiki/Square#/media/File:Regular_polygon_4_annotated.svghttps://en.wikipedia.org/wiki/Square#/media/File:Regular_polygon_4_annotated.svghttps://en.wikipedia.org/wiki/Square#/media/File:Regular_polygon_4_annotated.svghttps://en.wikipedia.org/wiki/Square#/media/File:Regular_polygon_4_annotated.svg

Triangular Numbers

resolve this square into two equilateral triangles

[west bronze]

I learned a lot from this thread that I did not know before. I wrongly knew that two consecutive triangular numbers make a square until I corrected my thinking on this. And people in this thread helped me see the truth of the matter.

https://cdn.discordapp.com/attachments/1239744236316790835/1239777270256635984/image.png?ex=6653514f&is=6651ffcf&hm=47ee889cb0471c010b0089aa9c5dd9880dd02adbbeee95841a8e8040c3ce78a9&

This image helped me see exactly how they are conjoined to make a non-square number.

Non-square because they contain no right angle anywhere to be found.

https://cdn1.byjus.com/wp-content/uploads/2016/06/triangular-numbers.jpg

These are the shapes and numbers we are reviewing.

now a question has arisen about the diagonal.

There are two diagonals as someone else noted

one is a rational line and the other is an irrational

I mean the diagonals of a rhombus

[snow hamlet]

I regret mentioning this trivia...

[snow hamlet]

I wrongly knew that two consecutive triangular numbers make a square until I corrected my thinking on this
... since it might have pushed you to further fixation

[west bronze]

Any time that rafain would like to present to us a rhombus with a 60 degree angle and a square both situated on the same base and having the same area, we would greatly benefit in this conversation.

[snow hamlet]

I have already presented such a rhombus, which contains exactly the same amount of dots as the corresponding square configuration

These two configurations represent the same square number

[west bronze]

having the same base so as to assure that the units are uniform within the figure and from figure to figure if he should choose to make another accurate set depicting the situation correctly

[snow hamlet]

No, units are irrelevant to the numbers

[west bronze]

That would be a grand & glorious achievement for all of us

[snow hamlet]

Nor should correctness of math be determined by an uneducated person

[west bronze]

i believe it would send us into ecstasy

[snow hamlet]

Glory is subjective

[snow hamlet]

Believe as you will

[west bronze]

better than ecstasy

[snow hamlet]

Again, to you your own

[west bronze]

now that we have it fully understood that triangular numbers never produce square numbers due to a lack of a right angle

i am wondering what else is significant about these numbers

[snow hamlet]

No, the sum of consecutive triangular numbers is a square number

[west bronze]

because they once held great prominence in geometry and math

[snow hamlet]

You only know to wonder and not to disregard irrelevant facts

[west bronze]

they hardly get a mention in modern math

and I wonder why?

[snow hamlet]

They still do

[west bronze]

perhaps because they are not so easy to work with?

[snow hamlet]

Wdym, I used them a few days ago at my workplace

[west bronze]

like the square they are equilateral

[snow hamlet]

Are you referring to yourself, the one not easy to work with?

[snow hamlet]

Only during their derivation

[west bronze]

and there is something else interesting if we refer to Nicomachus further

[snow hamlet]

Did you refer to Nicomachus for your mention of "units"?

Or was that wishful thinking?

[west bronze]

in that they are produced through the additions of gnomons

[snow hamlet]

Would you clarify what the last word means?

[west bronze]

seems that figures were multiplied proportionately by this addition

[snow hamlet]

Where was multiplication involved?

And by what addition?

[west bronze]

and the triangular number is produced by direct addition of each successive number in the number series.

[snow hamlet]

Sequence, not series

[west bronze]

while square have an easily understandable relation between their number or area and the side

triangulars don't

[snow hamlet]

Again, areas and side lengths are irrelevant

[west bronze]

it's a tad bit confusing sometimes

but fortunately Rafain has supplied us with one good figure of use to us

[snow hamlet]

That's a direct consequence of not understanding definitions

[snow hamlet]

And you decided to misinterpret it to your own detriment

[west bronze]

because the addition of consecutive triangular numbers ifs correctly depicted in the one figure

[snow hamlet]

In both

[west bronze]

his second figure is worthless

[snow hamlet]

Says the uninitiated

[west bronze]

his second figure does not depict the construction of a square from two consecutive equilateral triangles or triangular numbers.

[snow hamlet]

When it really does

[west bronze]

his second figure can only be resolved into two right triangles

[snow hamlet]

No one fixates on the shape of the triangles like you do

Thought the topic was triangular numbers

[west bronze]

and therefore is not a correct demonstration of the construction of a square from two equilateral triangles

[snow hamlet]

Again, says the uninitiated

[west bronze]

i do hope that we will see him again and he will repent of his wickedness

https://cdn.discordapp.com/attachments/1239744236316790835/1239777828501717084/image.png?ex=665351d4&is=66520054&hm=03788685b79c52caead8a7d00e3521c5157a623720dd1f35856fe0d990fe1879&

[snow hamlet]

Does subjective belief suffice for your philosophical inquiry?

[west bronze]

This is the offending square

[snow hamlet]

Offended who?

[west bronze]

which contains no equilateral triangles and is odious to my sight

[snow hamlet]

To your sight, ah I see

[west bronze]

I do wish the best for him

and hope he gets well

[snow hamlet]

The best for Rafain would be the student to start learning

[west bronze]

and stops his trolling of this topic as Ted (AKA GARRY) did

Garry was honorable enough to own up

[snow hamlet]

Who trolled though

[west bronze]

I have little hope for the incorrigible

[snow hamlet]

Ironic

[west bronze]

Refain should simply examine the facts of his construction

https://cdn.discordapp.com/attachments/1239744236316790835/1239777828501717084/image.png?ex=665351d4&is=66520054&hm=03788685b79c52caead8a7d00e3521c5157a623720dd1f35856fe0d990fe1879&

there it is

no equilateral triangles anywhere in it

[snow hamlet]

Yup, which fact did I miss?

[west bronze]

and pure nonsense

[snow hamlet]

Don't care, next fact

[west bronze]

let's hope that he recovers from his problem with geometry and mathematics

[snow hamlet]

Numbers agree, angles do not, everyone in this discussion except you is fine with that

[west bronze]

he will always be welcome when he can cooperate and talk sense.

[snow hamlet]

Sadly this explains why you are not welcome

[west bronze]

triangular numbers? Anyone know anything about them?

Have you tried doing any math with these little buggers?

[snow hamlet]

.
1 + 3 = 4
3 + 6 = 9
6 + 10 = 16
10 + 15 = 25

[west bronze]

1 + 3 = 4
3 + 6 = 9
6 + 10 = 16
10 + 15 = 25

note that the addition of two consecutive triangular numbers produces a rhombus of 60 and 120 degrees and never makes a square!

[snow hamlet]

Who asked

[west bronze]

now that is something new and imaginative

[snow hamlet]

Irrelevant as well

[west bronze]

I am always inquiring when studying math

[snow hamlet]

New, not so much, since you have already repeated it so many times

[snow hamlet]

Have you inquired why the geometric properties matter to triangular numbers after derivation?

[potent hazel]

Except when they do form a square

[west bronze]

I am always asking questions of the books I read

I like reading both classical math and sciences

[snow hamlet]

And not of your own assumptions? That's a little hypocritical

[west bronze]

i like them very much

i can argue with all of their foolishnesses and they don't argue back

galileo

he's one that is interesting but i have a problem with one of his figures

descartes

i like reading math in the context of it's history

these triangular numbers attract me because they are no longer used in math for much

if at all

[snow hamlet]

So you thought

[west bronze]

and i wonder what caused their extinction

or what caused them to fall into disuse?

as i said

[snow hamlet]

Triangular numbers are the basis of Gauss sums, which are still very much alive for sum-of-digits interest allocation

[west bronze]

they become difficult if you are trying to work with the area and the side

there's no very clear correlation between the side and area

[snow hamlet]

So you are making it difficult for yourself?

[west bronze]

not like with square numbers

[snow hamlet]

There is, but they are irrelevant to the numbers themselves regardless

[west bronze]

still, we have to memorize our multiplication table and memorize the squares

who bothers to memorize and work with triangular numbers?

pascal includes triangular numbers prominently in his triangle

and i don't think he features square numbers

let's look and see

let's take a look at pascal

here he lays out his triangle thus:

his first row is all 1's

[small tulip]

who gon spend his time on studying these triangular numbers bro

[west bronze]

his next row is 1 2 3 4 5 6 7 8 9 and those form superparticular relations

which are important in harmonics

his next row features triangular numbers 1 3 6 10 15 21 28 36

the addition of consecutives makes rhombuses as noted

[small tulip]

i know whats the square root of the number 2

🥶🔥 too cold

[west bronze]

36 is know as a square but not in this context

it is a rhombus in itself needing no addition of two consecutives

it is not a square number here

[small tulip]

36 equals to 6 x 6 🔥

[west bronze]

what is unusual for 36 is that it needs no addition to make it a "square"

there is no addition of consecutive required to produce it

[west bronze]

yes but in this system we would haev a rhombus with a 60 degree angle

so keep squares out of this

[small tulip]

i suck at geometry bro

[west bronze]

the 3rd row looks like this:

[small tulip]

chemistry better than geometry trust

[west bronze]

1 4 10 20 35 56 84

he has some other weird rows

[small tulip]

4 is a square

[west bronze]

there is no progression of squares in pascal's triangle!

[small tulip]

yeah but 4 is a square bro

[west bronze]

well 4 would be a square yes

[small tulip]

2 x 2

[west bronze]

as long as it is not produced from the addition of two consecutive triangular numbers

[small tulip]

aaa

[west bronze]

right 2 x 2 = 4 and 4 is square

[small tulip]

real

[west bronze]

but 1 + 3 is not square because these are two triangular numbers

likewise

3 x 3 = 9 and that is square

[small tulip]

6 is a triangular number bro

[west bronze]

but we don't have that with triangulars

3 + 6 = 9 and is not square

[small tulip]

9 isnt a triangular number

[west bronze]

because it is not produced by two equal side 3 x 3 = 9

[small tulip]

9 is a square

[west bronze]

instead it is produced by 3 + 6 = 9 and is non-square

[small tulip]

hollon 3 plus 6

[west bronze]

9 is square only when it is produced by 3 x 3

[small tulip]

the square root of the number 9 is 3

[west bronze]

not if it is produced by 3 + 6

there is no square root of a triangular number

and triangular numbers ever produce square numbers

[small tulip]

so the formula its gonna be uhh

[west bronze]

in the series 1 3 6 10 15 21 28 36

[small tulip]

how do i put the square root thing

[west bronze]

36 is a triangular number and a 60 degree rhombus

[small tulip]

$

[west bronze]

I think there is a tri root

[small tulip]

nah

[west bronze]

it doesn't make much sense though

[small tulip]

the square root of the number

plus

that square root

multiplied by 2

[west bronze]

how can a triangular number have a square root?

[small tulip]

an example is 9= 3+6

[west bronze]

it must have a triangular root on 3

not a square root

[small tulip]

boring imo

[west bronze]

numbers can mean different things

i don't find it boring at all

[small tulip]

pizza slices

[snow hamlet]

Yes! Yes! Yes!

[small tulip]

but yeah

is 9 a triangular number

[snow hamlet]

The same number can be the number of dots in a rhombus configuration, as well as the number of dots in a square configuration!

[west bronze]

i find it to be a very exhilarating experience with mathematics

no 9 is not a triangular number it is a rhombus

[small tulip]

hollon

[west bronze]

because it is produced through the addition of two consecutive equilateral triangles

[small tulip]

there are numbers that represent geometrical figures?!

[west bronze]

sure

[small tulip]

i mean

[west bronze]

I posted Nicomachus here before

[snow hamlet]

Rather, there are geometrical figures which represent such numbers

[small tulip]

currently im doing trigonometry at school

and i cheat in all of my exams

[snow hamlet]

Not exactly the same topic

[small tulip]

but bro

havent heard of triangular numbers till now

[small tulip]

are there numbers that represent rhombuses?

[west bronze]

here it is from book 2

[small tulip]

ay bro i was right

thats why theyre called square roots bro

[west bronze]

according to pascal's treatises there are triangular and pyramidal numbers

[small tulip]

because they represent squares!

[west bronze]

right

square roots represent squares

we are in another world of math here

there are no squares

[small tulip]

triangles

thats why theyre called triangular numbers

cuz they represent a triangle

[west bronze]

pascal's triangle (misrepresented on this very forum) is not made up of all triangular numbers

[small tulip]

took me long enough to figure out

[west bronze]

but it does feature them prominently and i think the pyramids also

but there are no square numbers

because they don's exist here

[small tulip]

hollon hollon

so

a triangular number

[west bronze]

look

[small tulip]

i mean 8

[west bronze]

it is just a convention

[small tulip]

its gonna be

[west bronze]

you were taught conventions in math

[small tulip]

8 plus 7 plus 6 plus 5 plus 4 plus 3 plus 2 plus 1

[west bronze]

along comes a nutball who disregards those conventions

[small tulip]

what is it

[west bronze]

and examines these numbers in another context

[small tulip]

whats the context

[west bronze]

and they make another logical sense that is completely at odds with your established conventions

but I am speaking from a math that is rooted in history

the context is the history of math

[small tulip]

hollon but the triangular's numbers triangles are equilateral?

[west bronze]

yes

[small tulip]

aaaaaa

[west bronze]

and they have nothing to do with right angles

[small tulip]

i got it now bro

the formula is

[west bronze]

and it's very very interesting that there was a discussion here about why we have 360 degrees in a circle

and 36 happens to be a triangular number and a rhombus

[small tulip]

x + x-1 + x-2 + x-3 + ... + x-n

trust

[small tulip]

are there numbers that represent rectangles?

[west bronze]

this is the point i have been trying to get to for weeks with the opponents of this system

a square is a rectangle

[small tulip]

no bro

[west bronze]

you already have the answer to that question

6 x 6 = 36

[small tulip]

a

[west bronze]

that is a square because it is made of 6 x 6

[small tulip]

they both have 360 degrees in total

[west bronze]

4 x 9 is a rectangle

[small tulip]

yes

hollon

[west bronze]

there is a 90 degree angle

[small tulip]

is 9 x 16 a rectangle

[west bronze]

ok ok ok

do your own independent study

scroll back adn read what was written

i already went over this a zillion times

get it right and return

i am blocking you

nEXT

anyone else here want to learn about triangular numbers

because i am game for learning anything about these numbers

[snow hamlet]

Who misrepresented Pascal's triangle?

[west bronze]

as long as the participants maintain the context of the discussion

the context is triangular numbers and squares do not exist.

square do exist in some other system but not in this

and that is probably why they fell into disuse

probably because people could not comprehend it when confronted with it

probably wars broke out between cultures

and we haev our contemporary math because some oen won the war

not for any other reason

the logic of triangular numbers is just as sound as any other math

people get stuck in their thinking

and cannot imagine another world of math

i don't know how people can do math for modern physics and not understand that this is simply a different system of math

our contemporary system of math causes the dysfunction of other maths when encountered

triangular numbers do not really function well in our contemporary system

we have seen repeatedly that people just refuse to comprehend what has been said

it is not that they are incapable but they adhere rigidly to a given convention

and cannot use their imaginative powers beyond it

the idea of number in relation to shapes is one of the most ancient relations in the history of ideas

you have simply been asked to attach familiar numbers to unfamiliar shapes

if you are used to seeing squares you will se squares

i have asked you to relate these numbers to triangles

and precisely equilateral triangles

[snow hamlet]

Related I have

[west bronze]

and every participant here to this date refuse to depart from the idea of right triangles

you do not belong in this discussion

[snow hamlet]

Just as you refuse to depart from the angles of the triangle representation

[west bronze]

this discussion belongs solely to those who can imagine a triangular system

[snow hamlet]

which serve no purpose to the numbers themselves

[snow hamlet]

Who said so

[west bronze]

and the introduction of any talk of right triangles or squares is out of context

can you make an equal sides triangle fro right angles and squares?

NO

[snow hamlet]

Then delete the page from Nicomachus mentioning square numbers

[west bronze]

you cannot

[snow hamlet]

Why not

[west bronze]

can you resolve squares into equal sided triangles

no

you can not

therefore you are out of place to enter them into the discussion

[snow hamlet]

No, you cannot, we can dilate the triangles at will

[west bronze]

they are wholly irrelevant

[snow hamlet]

So you think

[west bronze]

having nothing to do with triangular numbers

[snow hamlet]

Having nothing to do with the equilateral triangles, yes, but relevant to the triangular numbers all the same

Since the number of dots / circles in the right-angled configuration is still the same

That is the triangular number

[west bronze]

so go away and come back when you are willing to talk about triangular numbers and the numbers and shapes that they generate.

so now

[snow hamlet]

You are reversing the order

The shape generates the number

The shape is thus spent after this generation

and no longer the point of contention

[west bronze]

in our system we define 1 4 9 16 25... as non-square numbers

[snow hamlet]

Who would use this system besides you?

[west bronze]

and we define them as containing a 60 degree angle and a 120 degree angle

[snow hamlet]

Numbers do not keep track of the angles of triangles generating them

[west bronze]

and at no time does a 90 degree angle make appearance until we work it out

[snow hamlet]

They have but one value each

[west bronze]

we will not admit squares or right triangles into this system directly

[snow hamlet]

We did not either

We just added numbers

[west bronze]

if they enter in at all they must come into the system indirectly

[snow hamlet]

The figures were just for ease of comprehension, for your meager imagination

[west bronze]

and they will always be subordinate objects of thought

they are not allowed to define our system of math

[snow hamlet]

Did you see Omegabet_ directly adding numbers?

[west bronze]

the definitions have been laid down

[snow hamlet]

#1239744236316790835 message

[snow hamlet]

Your definition has no binding on what numbers are

They, being real numbers, still get to be added

[west bronze]

so if we should meet again and i ask you about triangular numbers you will speak about equal sided triangles and not about right triangles or the figures that they generate

that's what I want to explore in this topic

[snow hamlet]

I am speaking that at this very moment

As in, the numbers are generated by equilateral triangular configurations

and the configurations are not relevant once the numbers have been generated

[west bronze]

Now a right triangle has made an appearance in this discussion from the demonstrations of a few

[snow hamlet]

Now who's fixated on that again?

[west bronze]

and they proved to be irrational

having no relation to our numbers or figures

[snow hamlet]

What does an irrational right triangle even mean?

[snow hamlet]

They are related to the numbers all the same

We don't need to discuss that anymore though

[west bronze]

that goes for any demonstration that introduces right angles

[snow hamlet]

It's your own loss, we don't necessarily need them though

[west bronze]

we want to explore the properties of triangular numbers

[snow hamlet]

One being that every consecutive pair adds to a square number

Note that I only mentioned square numbers, without mentioning any squares

[west bronze]

is it a valid argument to say "yes, but all equilateral triangles resolve into right angled triangles"

what can we say to this

[snow hamlet]

Yes, but we don't have to discuss that either

[potent hazel]

that's not an argument, that's just a fact

[west bronze]

someone (Ted aka Garry) tried this

[potent hazel]

yes

[snow hamlet]

He erroneously considered areas

[potent hazel]

and we already discussed they did a mistake

[snow hamlet]

That was his error

[small tulip]

aint triangular numbers like the base's "circles" added with themselves till 1?

[west bronze]

find his work and bring it up for examination

[potent hazel]

you really like being useless dont you?

[small tulip]

i understand the concept but i dunno how to explain it

[potent hazel]

we've all agreed that is the one error'd idea

[west bronze]

yes it is a fact that all equilateral triangles resolve to right triangles

[potent hazel]

triangular numbers are defined by 1+2+...+n for each natural n. Ie the number of dots needed to form a triangle

[west bronze]

now this was the assumption that served as a given for two poorly constructed proofs which fail

[small tulip]

i understand that

[west bronze]

go back and examine the work that was already done

and report the cause of failure

[small tulip]

is there a formula for that btw

[snow hamlet]

n * (n+1) / 2

[west bronze]

begin with Ted's demonstration

[small tulip]

thanks

hollon isnt n x (n+1) / 2 gauss' formula or am i dumb

[west bronze]

then go on to Rafain's demonstration

and show us the flaws in reasoning

[potent hazel]

yes

[small tulip]

yea bro

i studied that when i was 11

why would i repeat that bro ☠️

[potent hazel]

$S=1+2+...+n\implies 2S=(1+n)+(2+n-1)+...+(n+1)=n(n+1)$

[brittle loom]

Omegabet_

[west bronze]

none of that fancy talk will help you out

it's just gibberish

look up Ted's demo and cite it's problems

look up Rafains demo and report it's poor construction

it's not the whole demo that is flawed

the first figure fits very well wit this discussion

the second figure greatly distorts what is being discussed.

[small tulip]

uhh sorry for the argument changing but what are you tryna say to me

what are you tryna demonstrate to me

[west bronze]

the problem that you are experiencing here is that you are trying to get your right triangles into this discussion and they just dont fit

[potent hazel]

you asked what triangular numbers were, then asked about the formula

[west bronze]

and without those you cannot compose a square

ask Ted

he tried to introduce right angled triangles into the discussion in his demo

and it failed

you dont see him here anymore do you?

because his demo failed

[small tulip]

i dont

i mean

right triangles dont fit in these triangular numbers

[west bronze]

he could not successfully introduce a right triangle into this discussion

[small tulip]

only equilateral triangles do

[west bronze]

for one good reason

[small tulip]

i understand that

[west bronze]

you have to derive that right triangle from the equilateral

and while you can get that far

[snow hamlet]

Why don't they?

[west bronze]

you cannot construct a square with that particular right triangle

now just give up and go away

[small tulip]

i dunno bro

[west bronze]

because this is the way it stands

[small tulip]

i give up on this gang 🗣

[west bronze]

that demo failed

it was a very noble effort

i appreciate the work that Ted did

i think it was even nobler to admit that he was wrong

that is much better than you have done

I will rate the perfoemances of each of you

so far Ted gets the highest honoyrs

simply because he saw that he was wrong and shut up

[snow hamlet]

Who are you to rate anyone?

[west bronze]

now you might want to learn something when you are in my presence instead of clogging up the discussion for hours with no notable mathematics

[small tulip]

i still know gauss' formula

[potent hazel]

that's more math than rock knows

[small tulip]

yeah bro

[small tulip]

but wait

you know those right angled triangles

[west bronze]

anybody got anything on the topic?

[small tulip]

they can make numbers aswell

like 6

yea i give up on this

[west bronze]

yer not stoopid

you should give up on it

and come back when you can talk about triangular numbers

thank you for giving up

Let's get some people in here who want to learn

as i do

[potent hazel]

you dont want to learn though

you want to argue/find people that agree with you

[west bronze]

I want to learn about triangular numbers

[potent hazel]

An integer is a triangular number if that number of dots can form a triangle

[west bronze]

i do not want to talk about you or me

[potent hazel]

or, more properly, $T_n:=1+2+...+n$

[brittle loom]

Omegabet_

[west bronze]

i want to talk about and learn more about triangular numbers

[potent hazel]

yes

nowhere did i talk about you

I defined 2 (equivalent) notions of triangular numbers

[west bronze]

so if anyone can make some constructive contributions thatis welcome

[potent hazel]

I just made 2

[west bronze]

ok I'll unblock you for 2 minutes

now what do you want to say about this formula?

[potent hazel]

that's what a triangular number is

[west bronze]

yes

i know

[potent hazel]

you dont

but sure

[west bronze]

no

[potent hazel]

you're right, you dont

[west bronze]

i knew this before

[potent hazel]

anyway

[west bronze]

it's in the books

i could not quote it like you do

[potent hazel]

Ok, so you know $T_n=\frac{n(n+1)}{2}$

[brittle loom]

Omegabet_

[west bronze]

but i can certainly calculate a triangular numner

[potent hazel]

I dont think you can

else you wouldnt be arguing 9 isnt a square number

[west bronze]

i do not want to talk about mme, what i do or dont know, about you or anything else

talk triangular numbers

[potent hazel]

I have been

you keep lying to yourself and me

[west bronze]

so we have the formal formula

i dont really need that

i always work with numbers tat are easily at hand

ok i am blocking you

[potent hazel]

we are working with numbers tat are easily at hand

we're working with integers lol

[west bronze]

i will not accept ad hominem attacks in my discussions

[potent hazel]

not even integers, just natural numbers

[west bronze]

ok

keep talking

at all times i prefer to plug simple digits into a formula

due to my math illiteracy

so i do not need that formula

[potent hazel]

at least you accept your illiteracy now

[west bronze]

because everything that can be known can be understood in terms of low numbers like 3 6 10 15

[potent hazel]

Well not everything apparently

[west bronze]

i have always accepted the fact that i am extremely ignorant

[potent hazel]

since 3+6=9 cant be understood

[west bronze]

i also accept this about your level of ignorance

3+6=9 cant be understood

as the sum of two consecutive equilateral triangles

[potent hazel]

is 3 a triangular number?

[west bronze]

3 x 3 can be understood as a square 9

[potent hazel]

yes or no

is 6 a triangular number, yes or no?

[west bronze]

the triangulars are 1 3 6 10

[potent hazel]

is 9 a square number, yes or no?

[west bronze]

you just add the numbers up

[potent hazel]

yes or no

[west bronze]

you can also do this with gnomons

[potent hazel]

are 3 and 6 triangular numbers, and is 9 a square number?

[west bronze]

no

that satement is false

[potent hazel]

which part?

I assume 9 being a square number?

[west bronze]

3 x 3 = 9 is a square is true

[potent hazel]

yes

[west bronze]

3 + 6 = 9 is a square is false

ok

block

[potent hazel]

is 9 a square number?

[west bronze]

i am not repeating myself again

that is all

goodbye

[potent hazel]

I know you're upset about not being blindly believed

and actually having to think for once

but if you want to have discussions, come into them ready to be wrong tbh

[west bronze]

9 is a square number if and only if it is the product of an equal pair of numbers in this case 3

it is not a square when it is the sum of two consecutive eqiuilateral triangles

GOODBYE

[small tulip]

sorry but thanks to @snow hamlet i understood the concept more bro

those right triangles werent useless at all

thanks again @snow hamlet

[potent hazel]

he left again

[small tulip]

lmao i was right bro

ez thats because of god bro

[snow hamlet]

I believe that deep down, we believe in the same substance

• that 3 and 6 are consecutive triangular numbers that add up to 9, a perfect square
• that the combination of equilateral triangular configurations at unit spacing of 3 objects, and of 6 objects, do not form a square configuration of objects under rigid motion on the plane; instead they form a rhombus configuration

I don't know why rockhoven is so reluctant to refine his terminology and keeps having us change lingo and negate the first statement, at least in name

[small tulip]

yeah bro he told me to give up but somehow i know thales' theorem

[snow hamlet]

Of course they weren't, they were drawn with consecutive triangular numbers in mind

[small tulip]

and i brought squares as well

[west bronze]

that is an iscoseles right triangle

you are off topic

[small tulip]

but thats a triangle

thats still a triangle

[west bronze]

the topic is not isosceles right triangles

we are not taking about any triangle

you have a square

divided into teo consecutive equilatera triangles

[small tulip]

hollon bro so youre talking about equilateral triangles

bro

[west bronze]

you dont\because you cant

so go away

[small tulip]

lmao

yeah but hollon

[snow hamlet]

@west bronze Please read this for a moment, and think about it

[small tulip]

6 can be a triangular number in a right triangle

[west bronze]

so let's look at the kinds of shapes and numbers that triangulars can produce

we know that they produce rhombuses with 60 degrees

what about the pyramids?

[small tulip]

its the number of "circles" that matter bro

[west bronze]

that might be an interesting place to go

[small tulip]

not the shape

[west bronze]

I find it quite interesting that pascal totally ignores square numbers

the only reason that 36 appears is because it is a triangular

so 36 can be a triangular or a square because it can have equal sides

but 36 can also be a rhombus if it is constructed of two consecutive equilaterals

[snow hamlet]

This kinda isn't true since Pascal's triangle obviously lists every natural number, but I get what you mean

[west bronze]

I need a symbol that represents two consecutive equilaterals

we can use 2

[small tulip]

and 4 right triangles 🔥

[west bronze]

only in the context of squares

pascal is referring to 36 as a triangular number

not a rhombus or square

[small tulip]

i know

a right triangle is still a triangle

[west bronze]

it's not a triangular number because it is not equilateral

[small tulip]

i totally know what are you on right now

but bro you still talking about the "EQUILATERAL" triangle

[west bronze]

so it's no good to pass off right triangles as representing 2c=

2c= is my notation for two consecutive equilateral triangles

Triangular Numbers meaning: an equilateral triangular number

2c=tri do not add up to a square

[small tulip]

you shouldve specified

[west bronze]

now it is specified

[small tulip]

good

[west bronze]

it was specified repeatedly for the past weeks

you might like to scroll up and review the thread

[small tulip]

in the past weeks i wasnt here 🗣

but yeah i know what you on

[west bronze]

taking special care when viewing the demonstartions against the proposition

now we have another.

in the last figure posted

what type of triangle is that?

[small tulip]

uhh what figure

[west bronze]

https://cdn.discordapp.com/attachments/1239744236316790835/1244040860560134174/image-6.png?ex=6653aad6&is=66525956&hm=89300e508021c9350ef544ce671c8c4136b009da1533402336003cefb3977ddb&

[small tulip]

yes

[west bronze]

what kind of triangle is this?

[small tulip]

this is a straight triangle

[west bronze]

a right

[small tulip]

isosceles one

[west bronze]

yes

[small tulip]

hollon

[west bronze]

so what is the point of this presentation of a right iscoceles triangle?

that is what I would like to know

[small tulip]

its the same as a equilateral one

because

it has 6 dots

and the equilateral one has 6 dots as well

[west bronze]

an equilateral is equal to a right triangle?

[small tulip]

in terms of dots

yeah

[west bronze]

are the areas equal?

[small tulip]

we are NOT talking about areas

[west bronze]

we are not talking about right triangles

[small tulip]

were talking about "triangular numbers"

[west bronze]

why dont we talk about equilaterals?

[snow hamlet]

He didn't read my message huh

[small tulip]

dont bring angles bro

[west bronze]

right

what is a triangular number but an equilateral?

it is fully defined by Nicomachus

[small tulip]

whos that bro ☠️

[snow hamlet]

1 + 2 + ... + n, as indicated by the configuration?

[west bronze]

search this thread for nicomachus and pull up the page

I'm not doing all your work

[small tulip]

1 + 2 + ... + n = n x (n+1) / 2

i bet this the formula

(mentioned before)

[west bronze]

what does a right iscoeles triangle have to do with this discussion about equilateral triangles?

[snow hamlet]

He will reject the "formal formula" in no time

[small tulip]

to demonstrate that THE NUMBER OF DOTS MATTER

AND NOT THE SHAPE

[west bronze]

does that formula produce right triangles?

[small tulip]

YOU CAN HAVE A BUCKET FULL OF APPLED AND AT THE SAME TIME YOU CAN HAVE A BAG FULL OF APPLED

[west bronze]

but that is not the definition of a triangular number

[small tulip]

it is

[west bronze]

its an equilateral number

[snow hamlet]

No, it produces the number of dots in the equilateral configuration - not that the angles matter

[west bronze]

look at the topic title

now what are we discussing?

[small tulip]

bro

[west bronze]

equilateral triangular numbers

why does everyone insist upon derailing this topic into other types f triangles?

[small tulip]

were discussing the fact that equilateral triangles are the same as right triangles IN TERMS OF DOTS

not area

not perimeter

[west bronze]

let them be so

[small tulip]

THEY ARE

[west bronze]

they dont have the same angles

[small tulip]

angles

bro

[west bronze]

the numbers are derived form the shapes in this theory

[small tulip]

its the same

[west bronze]

if you dont like the theory i cant help you

this is the number theory of the ancients

[small tulip]

bro

you can deny everything you want

but in terms of dots

[west bronze]

dots

[small tulip]

a right triangle is the same as an equilateral one

[west bronze]

you can have the same dots

and we would have the saem number

[small tulip]

we DO have the same number

[west bronze]

if the dots are spaced equally from one example to teh next

yes

the demo here was not done so

[small tulip]

THEY ARE

[snow hamlet]

@small tulip Could you refer rockhoven to my message above?

[small tulip]

sure

[west bronze]

the spaces between the dots are distorted to give an incorrect perception of the proposition and the refutation

[snow hamlet]

Reply to this one

[small tulip]

@west bronze

[west bronze]

I am going to do the demo correctly when i have time

[small tulip]

do you have a formula for allat

js say yes or no

[west bronze]

no

[small tulip]

then how do you expect to calculate the number of dots

manually?

[west bronze]

all we need are the dots and equal spaces between the dots

then everything will be much more clear

[small tulip]

bro

[snow hamlet]

All we need are the dots really

[small tulip]

the space between the dots does NOT have to do with this topic

if you have 3 dots

you can build a equilateral triangle ofc

and

[west bronze]

what would work best is to have a grid

[small tulip]

a right isosceles one

[west bronze]

with equally spaced units

[small tulip]

they are equally spaced

the dots ARE

[west bronze]

and on each unit would stand both a square and an equilateral triangle

[small tulip]

why u bringing squares now

[west bronze]

then we can make a very clear presentation of the facts

and area does matter

and we will see so in such a demo

[small tulip]

do you have a formula for areas?

say yes or no

[west bronze]

yes or no

[small tulip]

i think thats a yes

thats logical

[west bronze]

yes or no

[small tulip]

so if we have an area formula

[west bronze]

no

[small tulip]

then why cant we have a formula for a triangular number

[west bronze]

you guys have already distored areas repeatedly

[small tulip]

bro

[west bronze]

i dont trust you to handle any formula to calculate these areas

[small tulip]

we all know that the areas are different

but bro

[west bronze]

your whole mission here is to distort the meaning and significance of the topic

[small tulip]

the dots remain the same

[west bronze]

you are not here to discuss the topic

you are here to prove that i am wrong

[small tulip]

bro

[small tulip]

all im saying is

[west bronze]

you have no real interest in triangular numbers

[snow hamlet]

Talk about ad hominem

[small tulip]

do you have a formula to calculate the number of dots?

in a equilateral triangle

you still aint answering bro

[west bronze]

https://cdn.discordapp.com/attachments/1226524052420689921/1243752286841208963/PascalTriangulo.jpg?ex=665346d4&is=6651f554&hm=496679b871fc17d244fea94f8248c554d65794ecc6902f8e1bbfa22ab417b618&

[small tulip]

lets take it easy

[small tulip]

thats a right triangle

isosceles one

[west bronze]

square numbers do begin to appear in this table

but not immediately

[snow hamlet]

That's Pascal triangle, so not exactly the same thing

[west bronze]

by the end of his demo

we see only a few squares

4 and 9

[small tulip]

i know

and 36

if im not wrong

[snow hamlet]

and in fact, 16 and 25

[west bronze]

and 36 but this number appears as a successor in a series of triangulars making 36 triangular

[snow hamlet]

since the 2nd row and 2nd column keep going on

[west bronze]

and they appear rather late

it's a peculiar approach to math isnt it?

[small tulip]

it is

[snow hamlet]

What is peculiar about it?

[west bronze]

and then he only seems to build pyramids out of these

[snow hamlet]

Who built pyramids? What pyramids?

[small tulip]

hollon arent pyramids in 3d

not 2d

[snow hamlet]

Ah pyramidal numbers

[west bronze]

they appear in one of the rows

1, 4, 10, 20, 35, 56, 84

[small tulip]

aaaaaa

continue

[west bronze]

thats it

that is in the 4th row right after the triangulars

this is a very interesting diagram

because he seems to be referencing ancient mathematics

[snow hamlet]

We can apply the same logic for pyramidal numbers - the number of objects in right-angled pyramidal configuration is the same as equilateral pyramidal configuration

[west bronze]

he has a row of superparticulars

a row of triangulars

and a row of pyramidals

in that order

[small tulip]

i aint talking on this one since i dunno how to respond

[west bronze]

IDK either

i just think its super curious

[small tulip]

it is

[west bronze]

but he too uses squares and right angles isoceles triangles to map this

[small tulip]

i know

[west bronze]

i just think it's serves no purpose to do so

it is misleading

if we wrongly associate right with with equi

we can easily be led into the equally wrong idea that we can construct squares out of equilaterals

[small tulip]

its futile at this point proving that every triangle can associate with triangular numbers

[west bronze]

which wont work

the thing is this

[small tulip]

yea but the number of dots will remain the same lmao

[west bronze]

youse guys are in the habit of associating numbers with formulas and using coordinate geometry which relies heavily on right triangles

[small tulip]

formulas do help us

[west bronze]

the heart of the first problem solved with analytical geometry is an equilateral triangle

ok

[small tulip]

yea ig ill continue tomorrow

im sleepy now

[west bronze]

now i think of number in a classical sense

[small tulip]

its almost 1am bro

[west bronze]

associated with definite shapes that cannot be altered into other shapes so easily

[small tulip]

@snow hamlet can you continue please

[snow hamlet]

I would just like rockhoven to read my comment back there

[west bronze]

where?

[snow hamlet]

Here

[west bronze]

We can apply the same logic for pyramidal numbers - the number of objects in right-angled pyramidal configuration is the same as equilateral pyramidal configuration

i can agree to everything except refering to the two consecutives as a square

[snow hamlet]

The numbers really do not discern the configurations once their values are known

[west bronze]

the area is equal to a square number

[small tulip]

yea ill leave it to @snow hamlet

[snow hamlet]

No, the area is not

[west bronze]

but i am on the trail of other game

[snow hamlet]

Nope, Imma sleep as well

[west bronze]

and i find that terminology to be a huge distraction

[small tulip]

Good choice

cya @west bronze

see you tomorrow

@snow hamlet you aswell

[west bronze]

this is where we need an accurate demo

that will settle the dispute

maybe

[snow hamlet]

Which you are not able to produce, so why assert?

[west bronze]

i can produce it

[snow hamlet]

You do that then

[west bronze]

i have not the tools though

[snow hamlet]

I will sleep

[snow hamlet]

So you cannot

[west bronze]

i need a grid that has units expressed as squares adn as equilaterals

[snow hamlet]

Find one

[west bronze]

i dont know where to find such a grid

but you can imagine it

[snow hamlet]

Yes, you don't know

[west bronze]

just imagine a triangular grid of equilaterals

and each unit also is expressed as a square

[snow hamlet]

You assert a lot of stuff despite your ignorance
such as how the addition of triangular numbers must be done as the union of the corresponding equilateral triangles

[west bronze]

i may have to find one grid and draw the other over it

[west bronze]

this just comes from the books

[snow hamlet]

It does not

[west bronze]

that two consecutives add up to a square

[snow hamlet]

Where does the book tell you to "add up" shapes to get a sum?

[west bronze]

well you did so in your demo

[snow hamlet]

I didn't

[west bronze]

you added 3 and 6

[snow hamlet]

I added the count of circles

not the areas

[west bronze]

how did you generate it?

[snow hamlet]

nor union-ing any shapes

I added the counts of circles in the figures

This is just addition of numbers

[west bronze]

right

according to the formula

[snow hamlet]

of which the kindergartener is aware

[west bronze]

they are not

now i am going to turn you off

[snow hamlet]

They know how to add 3 to 6

[west bronze]

i want math spoken here

not bs

[snow hamlet]

Does this not make sense to you?

[west bronze]

goodbye

they do not

that is 1st grde

grade

[snow hamlet]

Fine, that may be true where you live

[west bronze]

fine

they do it in kinder

ok

what is the point

[snow hamlet]

It is just addition of numbers