#Define a Triangle

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sand ivy
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doesn't mean its right

elder gust
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I'll argue with anyone

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don't care if its Euclid or not

sand ivy
elder gust
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Now you say an equal side plane can be divided into two triangles

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yet those triangles are supposedly "right"

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and I am having a problem with the entire concept of a right triangle

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Exhibit A: the isosceles right triangle

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I can see an equilateral triangle

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after that I am in doubt

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and I am nearly certain that a right triangle is not a triangle at all

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I have to admit that it is a triangle according to Euclid's simple definition

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Since I first posed this question no one has jumped in to say

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"It's a trilateral figure"

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and that is probably because after defining a triangle

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Euclid went on to investigating their properties

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and knowing these properties after Euclid explained them

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we think of a triangle as more than just his simple definition

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So I think there is some unstated modern definition

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that would make more sense to us 2500 years after the fact

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and so much has been said about triangles

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much of it being questionable

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here's the thing

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if we accept his definition then the case is closed

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I cannot argue about the vertices being truly closed

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I cannot argue about whether an irrational line actually meets a rational line at a common point

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because his definition is "trilateral figure"

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But I don't think that we can accept that definition

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otherwise why did someone state it as "three non collinear points?"

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why am I thinking of closed vertices?

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Why are you thinking of the sum of two lines is greater than the remaining side?

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There is something unsatisfactory about Euclid's definition

sand ivy
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in geogebra

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need a diagram

elder gust
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in light of all the known properties of triangles

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Well geogebra is going to be set up according to convention

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it is not going to think the thing through

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it is only going to respond to a program

sand ivy
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๐Ÿ˜ญ

elder gust
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I could teach AI that a right isosceles triangle does not add up

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but it will forget everything I taught it in two minutes

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because it will revert to it's program

sand ivy
elder gust
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so that is what led to my investigations of the angles and sides

sand ivy
sand ivy
elder gust
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Why would we need a drawing of that?

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the 90 degree angle is subtended by an irrational line

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yet the 90 is commensurable with te 45

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that does not add up

sand ivy
elder gust
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something is rotten in denmark

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I've explained this two or three times

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did you not understand?

sand ivy
elder gust
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ok

sand ivy
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most i read the last text part

elder gust
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well youhave all the pieces there

sand ivy
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only there a lot of messages

sand ivy
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two right angles make an isolceous

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isosceles*

elder gust
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a triangle cannot have commensurate angles and incommensurate sides

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because sides are proportional to the angles that they subtend

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so I cannot accept the idea of a right triangle

sand ivy
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๐Ÿคฃ

elder gust
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it is not true

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math does not tell the truth

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i didnt say that i did not like it

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i like right triangles

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i wish that they existed

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they are quite useful items to have around

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this is an investigation into truTh

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with a capital T

sand ivy
elder gust
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so one of the properties of an isosceles right triangle is not a true property

sand ivy
elder gust
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why are you in this discussion?

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you don't seem to be following it

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you introduced the division of the square yesterday

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so i examined the results

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you don't seem to have followed the logic

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you proposed to divide a square into two triangles

sand ivy
elder gust
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I proved that this cannot be done

sand ivy
elder gust
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the properties of the triangles are false

sand ivy
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i am just watching here

elder gust
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the supposed triangles

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there is no such thing as a right isosceles triangle

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if it supposedly has 45, 45 and 90 degree angles

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all commensurate

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and two rational lines

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but one irrational

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this does not add up

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due to the fact that sides are proportional to the angles they subtend

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AGREED?

sand ivy
elder gust
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well, you did divide the square into two triangles according to Euclid

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but IDK

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we know so much more now

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the conception of a triangle seems much different nowadays

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I mean, look at the fact that no one

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I mean NO ONE offered Euclid's definition

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"trilateral figure"

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It is just too imprecise

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We have to gather a better Euclidean definition from his construction in 1.22

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Here he places them in position

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so we need that

sand ivy
elder gust
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then there is this notion of three non collinear points

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and three closed vertices

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and lines meeting at common points

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these are all modern properties of triangles

sand ivy
elder gust
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the thing is that since the evolution of the definition of triangles

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no one has bothered to examine whether the triangle fits the definition

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and this is a problem for me to communicate with other people

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because they think there is something wrong with the ROCK

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what has actually happened is that math has evolved so much that it is a predator unto itself

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classical and modern math are antagonistic to each other

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In page 5 of this article, the editors speak of a discontinuity between ancient and modern math

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This theme is discussed until the end of the article on page 11

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ancient and modern mathematics are not the same thing

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if there is one thing they hold in common

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it is the ambiguities, equivocations and outright errors that have been carried over

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into modern mathematics

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these are all willfully overlooked

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for a system that works

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if you are a modern you are a pragmatist

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E=mc2 is likely to be wrong

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like all theories

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but if you can blow everything up with it

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that is going to attract a lot of attention to your science

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and the truth of it will be immediately judge by whether it works

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that is pragmatism

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no one is going to listen to anyone trying to disprove the theory of relativity

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because you can't blow everything to smithereens with it

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it's an important lesson from the history of science

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that all theories eventually fall

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they all have to be updated

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math is an evolving science

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we are not working with truth

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but with theories

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The article asks several questions like:

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Are the modern innovations improvements or corruptions of the mathematical
arts and sciences?

elder gust
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The convenient way for my opponents to approach this

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and defend convention from my assault upon their mathematics

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is to strictly adhere to Euclid's original definition

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which only refers to sides or lines

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adapting that view I will be restricted from examining points, intersections, vertices and the commensurability of lines

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Who would be comfortable with that?

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"trilateral figure" - Euclid

sand ivy
elder gust
sand ivy
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Nothing is wrong

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Its worse

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Than the new one

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Thats how it goes

elder gust
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yep

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i agree

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copernicus is not wrong

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just in some details

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how are you with Euclid's definition?

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***from Book 1 Definition 19.
Rectilinear figures are those which are contained by straight lines, trilateral figures being those contained by three, quadrilateral those contained by four, and multilateral those contained by more than four straight lines.

Definition 20.
Of trilateral figures, an equilateral triangle is that which has its three sides equal, an isosceles triangle that which has two of its sides alone equal, and a scalene triangle that which has its three sides unequal.

Definition 21.
Further, of trilateral figures, a right-angled triangle is that which has a right angle, an obtuse-angled triangle that which has an obtuse angle, and an acute-angled triangle that which has its three angles acute.

Definition 22.
Of quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong that which is right-angled but not equilateral; a rhombus that which is equilateral but not right-angled; and a rhomboid that which has its opposite sides and angles equal to one another but is neither equilateral nor right-angled. And let quadrilaterals other than these be called trapezia.***

sand ivy
elder gust
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That might be going to far

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no definition is correct

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it's just a definition

sand ivy
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Like it talks on 2 dimension

elder gust
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he defines something for the purpose of investigating it's properties

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but it seems to me that in the course of the investigations

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the definition will be altered

sand ivy
sand ivy
elder gust
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it's the discovery of the properties of the defined object

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that changes it's definition

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he does not officially redefine the triangle in the course of the Elements

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but I think after Euclid most people thought differently about a triangle

sand ivy
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@elder gust isn't all triangle that isn't right angled made of 1 right angled triangle and 2 lines

elder gust
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Trigonometry says that we can find a point on a curve using the vertex of a triangle

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even with an irrational line?

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we can have a right angle with two lines

sand ivy
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Right angles yes

elder gust
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it's the closing of the triangle with a 3rd line that is complicated

sand ivy
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Is dependent

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On right angles

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2 ways of flipping shows

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Right angles

elder gust
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a mathematician will want precision

sand ivy
elder gust
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here's the thing about definitions

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we define it then we can investigate it

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by broadening the definition we can continue our investigations

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what I have said about the isosceles right triangle could ot have been said

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even after irrational lines were discovered

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which happened in the times of the Greeks

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First, the definition has to be refined

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The most comprehensive book in the Elements is Book X on irrational lines

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Yet, they never went back and examined the right triangle

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Specifically the isosceles right triangle

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this triangle does not make any sense with it's irrational side

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and it's commensurate angles

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you cannot have both

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who posted 1:1:โˆš2?

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was that you?

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in such a triangle we have 45, 45 and 90 degree angles

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yet the sides are incommensurate

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how can that be?

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I am not saying that the side must be double because the angle is double

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but it seems that they should at least be commensurate

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if an irrational subtends an angle

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i would expect that angle to be irrational also

sand ivy
sand ivy
elder gust
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that's a different proportion

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is that what you meant yesterday?

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I thought you posted 1:1:โˆš2?

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again, 30, 60 and 90 are commensurable

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I think the isosceles triangle illustrates my point sufficiently

sand ivy
sand ivy
elder gust
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With respect to โˆ C, the ratios of trigonometry are given as: SINE: Sine of an angle is defined as the ratio of the side opposite (perpendicular side) to that angle to the hypotenuse. COSINE: Cosine of an angle is defined as the ratio of the side adjacent to that angle to the hypotenuse.

sand ivy
toxic python
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A rectangle

elder gust
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This is only a scientific investigation. Try experimenting and reporting the results.

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That's where we are right now with some of the issues raised.

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But the main question needs no proof because definitions are conventions and cannot be proven.

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The definition must be in the historical literature.

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Who has the authority to define a triangle?

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At this point, we barely know who has previously defined it.

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We have Euclid and that is all.

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Is his definition acceptable to modern mathematicians?

still dove
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no

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take the jordan curve consisting of three lines between (a, b), (c, d), (e, f) which can be defined by a piecewise linear function.

then by the jordan curve theorem we get an interior and exterior for this triangle.

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theres a reason why people dont use euclid's axioms as a basis for geometry anymore

elder gust
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We don't use Euclid's axioms? I think we will continue to use his definition anyway. I don't see why you are introducing needless jargon into the conversation. This is always the strategy of people on this discord. Speak clearly so that anyone can understand you. If a triangle is three lines then just say so. Leave Jordan out of it. I am sure that he knows as little as anyone else. What is YOUR definition of a triangle? If you have none then produce some example from the historical record. At least with Euclid we know exactly what he says and can quote it.

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I think people are putting down Euclid on this discord because they don't know Euclidean theory.

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And that is a fundamental of mathematics.

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Euclid, Archimedes, Apollonius, Nicomachus, Ptolemy, Copernicus, Newton, Huygens.

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That's mat.

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I don't know what you people are talking about 99% of the time.

elder gust
# still dove no

And critique some of the ideas presented here. People have dropped off a few definitions.

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Do they work? Why? Why not?

still dove
elder gust
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OK

still dove
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like yeah theres a definition for a triangle as a polygon with 3 sides

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and most people are happy with this definition

elder gust
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What's the problem with Euclid's definition?

still dove
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and this definition can be made rigorous and analytic if you want

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oh wait im stupid

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yeah nobody has a problem with euclid's definition

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everyone is happy

elder gust
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no yer not stoopid

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i don't know

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that is why i ask

still dove
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like is that all

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

Triangle

A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called vertices, are zero-dimensional points while the sides connecting them, also called edges, are one-dimensional line segments. The triangle's interior is a two-dimensional region. Sometimes an arbitrary edge is chosen to be the...

elder gust
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Right.

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So what do you have to say to the couple of posters who say that a triangle is 3 non collinear points?

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The link you post cites a few other properties that are not in your briefer definition.

still dove
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they are probably just being brief

elder gust
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Points and vertices

still dove
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ok

elder gust
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So you did nt include those before

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no you do?

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yes you dont?

still dove
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thats implicit in the definition of a polygon

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id presume

elder gust
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should definitions be incomplete?

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it's very strange to think about

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I can't get over the fact that you have to know the properties in order to define

still dove
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i mean its not incomplete it just depends on another definition

elder gust
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yet the investigation uncovers more properties

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do these newly understood properties then belong to the definition?

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then definitions would need constant revision

still dove
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no?

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a definition is just a way of uniquely characterizing some object

elder gust
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I appreciate your input.

still dove
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the existence of some new discovered property doesnt make the old definition incorrect

elder gust
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But look at what I have just said about the definition and te properties

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What is the relation?

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Well, Euclid does not include all of these properties

still dove
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well properties are the things satisfied by objects obeying a definition

elder gust
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he just talks about sides

still dove
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ok

elder gust
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he says nothing about points or vertices

still dove
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ok

elder gust
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do we need those? why? why not?

still dove
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im fairly certain euclid's elements used points

elder gust
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and where did they come from?

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they aren't in the definition

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yes but he never really defines a point

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or a line clearly

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it's ll assumptions

still dove
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yeah cuz it was obvious to him what a point and line is

elder gust
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well it is not obvious

still dove
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how bout you define a point

elder gust
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ok

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IDK what a pointis

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i'll try

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historically, a point is discrete

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there are spaces between points

still dove
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yeah but those are properties of a point

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what is a point

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you can tell me a point is blue but what is a point

elder gust
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that's the question

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are the properties the components of the definition?

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I think so

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but

still dove
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ok

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so whats the definition of a point

elder gust
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how can he define it before he investigates it's properties

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it is a chicken and egg problem

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which comes first?

still dove
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whats the definition of a point

elder gust
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te definition or the investigation into properties?

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I have no idea

still dove
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the definition

elder gust
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if you want to bar the use of properties to define a point

still dove
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so why would euclid have any more of an idea

elder gust
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I am speaking from an historical POV

still dove
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ok

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but what is a point

elder gust
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Euclid defines a point as "that which has no part"

still dove
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ok

elder gust
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that's all I can say

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did anyone else define a point?

still dove
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https://en.wikipedia.org/wiki/Point_(geometry)

Point (geometry)

In geometry, a point is an abstract idealization of an exact position, without size, in physical space, or its generalization to other kinds of mathematical spaces. As zero-dimensional objects, points are usually taken to be the fundamental indivisible elements comprising the space, of which one-dimensional curves, two-dimensional surfaces, and ...

elder gust
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Someone previously defined the number 1 as a "thing exclusive of other things"

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the point and the number 1 are analogous

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if that is so

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then

still dove
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no they arent

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maybe that definition just sucks

elder gust
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what is your argument against that definition?

still dove
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the sun is a thing exclusive of other things

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treasure is a thing exclusive of other things

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a locked room is a thing exclusive of other things

elder gust
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I see no problem in visualizing the number 1 analogously as a point

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because geometry and arithmetic are expressions of each other

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this can be demonstrated from the historical literature.

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You have not presented a coherent argument against that poster's definition of the number 1

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Someone previously defined the number 1 as a "thing exclusive of other things"

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That was Metactal.

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@sand ivy Here is an idea

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Spose we say that in 45 45 90 that the lines carry those values

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then in 30 60 90 the same hold

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so what are the consequences?

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think about that one

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and let me know

sand ivy
elder gust
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the lines are not commensurate but the angles are

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if we give those commensurable values to the lines

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what follows?

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This could lead to some wacky math

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what do you think?

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Won't the irrationality of the diagonal be transferred to the units?

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So the lines will become rational

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but their units will be incommensurate with each other.

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this would greatly affect the interpretation of the Pythagorean theorem

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I think this is an important point to consider

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and will be useful when we return to the Triangular Numbers thread

sand ivy
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sin(30)

ABC = 90
ACB = 60
CAB = 30

sin(30) = BC/AC
tan(30) = BC/AB
cos(30) = AB/AC
@elder gust

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Solve it

sand ivy
elder gust
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Solve what?

still dove
elder gust
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I think the idea of the number 1 has evolved and it is causing problems. How do we define something that refuses to stay put? I certainly can't fault the poster. Nicomachus and Euclid, Archimedes and Apollonius would have agreed with him. The definition is only problematic for a modern. Why is that?.

still dove
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because all these people lived in ancient greece

elder gust
quiet lake
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Triangle is already defined

sand ivy
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@elder gust i got some findings

elder gust
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What do you come up with?

elder gust
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same goes for the point, line and any other figure

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@quiet lake what's the definition of a triangle?

quiet lake
# elder gust <@616165347552395271> what's the definition of a triangle?

In euclidean geometry: 3 points, let's call them A,B and C, connected with the 3 segments [AB],[BC],[AC].
In graph theory (a more general setting): a clique of size 3. When you apply this to the graph with it's vertices all the points of euclidean space and it's edges all the segments you get the definition of a triangle in euclidean geometry.

elder gust
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I have already quoted Euclid extensively and 3 points are not in his definition

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He does introduce points later in his investigations but not in his definition

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that is one of my questions

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how can you define an object before investigating

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you are defining a triangle post-investigation

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you are taking properties and adding them to the Euclidean definition

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I agree that points are involved

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but it's a misrepresentation of his definition to call it Euclidean

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by what authority can we define a triangle?

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what is the historical background?

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who, besides Euclid defined a triangle?

quiet lake
cerulean atlasBOT
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Djake3tooth

616165347552395271.png
quiet lake
quiet lake
elder gust
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hello

elder gust
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math and science are to be decided by the majority?

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LOL

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No you are really headed for trouble

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what is your definition of points?

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and if you use $\bR ^n$ you should explain it

cerulean atlasBOT
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rockhoven

699627666319605861.png
elder gust
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it is improper for people here to use equations without thoroughly explaining what they mean by it

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when using an equation provide some context

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that is why I call it gibberish

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A point is

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$\bR ^n$?

cerulean atlasBOT
#

rockhoven

699627666319605861.png
elder gust
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A relation squared?

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A point is supposed to be an indivisible unit

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it is indivisible because it has no dimension

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supposedly

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if it has no dimension, how can I draw a line through it?

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Oh I get it

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the line has no breadth

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yeah that one gets me too

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here in Euclid he says that a line is length without breadth

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OK

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the commentators say that Euclid defines his objects and then draws them in order to prove that they exist

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he cannot and does not draw a line without breadth

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so such object in fact do not exist

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This whole thing about 1 being divisible destroys the original concept of 1

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being an indivisible unit

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in that respect geometry mirrors this arithmetic in the indivisible point

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but if we are going to divide the unit 1

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why not go the next step and divide the point?

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$\bR ^n$

cerulean atlasBOT
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rockhoven

699627666319605861.png
elder gust
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What is that?

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Coordinate geometry?

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highly suspect

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a. Is R reflexive? b. Is R symmetric? c. Is R transitive? d. What is the meaning of the relation R^2? Specifically, when are two objects R^2 related?

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I do not appreciate how many of the posters in my discussions are navigating them

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There have been a large multitude of ideas posted in my topics

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rather than coming in here to engage each other and critique each other's ideas

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you are coming in to challenge me and my ideas

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exclusively

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you are not engaging with any of the other posters nor with their ideas

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metactal brought a definition of the number 1 worthy of much more attention

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engage him

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and challenge that idea

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his definition would hold for the point also

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a thing exclusive of other things

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that is what he said about the number 1

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and I agree that that is very close to the classical definition of 1

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1 cannot be divided without destruction of it's original definition

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it is the "final point" so to say

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in a classical sense, if we divide 1, we get 1

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because that is the final point

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an indivisible point

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and if we are going to divide the unit 1

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we might as well divide the unit point

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what would that look like?

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So now I think that I have demonstrated once again that math is full of ambiguities and equivocations

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when you can destroy the original meaning of the number 1

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and get away with it?

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you cannot divide 1

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if you insist upon dividing 1

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then I have to insist that you have infused 1 with the properties of other numbers

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so if you write 1/2

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you are only introducing the idea of 2 into 1

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and each one of those parts must be the unit

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so you have 1

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whatever way you dice up 1

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the result is 1

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under the classical meaning of 1

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as an indivisible number

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I'm all in for it

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as long as I am allowed to divide the point

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Now where was I?

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I want to examine the properties of certain right triangles

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and I have suspicions that they do not conform to definition

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@sand ivy has suggested examining the problem using tan, cos and sin

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I know little or nothing about that

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but go ahead because I am interested

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I just want to warn you that if the properties of right triangles are under investigation

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and if it is suspect that right triangles are not true triangles

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then it will not do to prove the truthfulness of right triangles

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by utilizing right triangles

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they are the objects under investigation

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the thing is this

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what are you searching for?

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are you searching for the truth +/or falsity of math?

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or are you searching to support conventions at all costs?

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I am searching the possibility or impossibility of truth in mathematics

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That is the nature of these investigations

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I care nothing for what the textbooks say

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or what your professors insist that you write on an exam

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I am not here to get a quick answer

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so that I can go happily on my way to that $50,000 yearly salary

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So if you have nblindly accepted the "truth" of mathematics

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before investigating whether math is true or not

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you are not going to be comfortable in my conversations

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and when we join together for conversation

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it is a cooperative effort

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we are discussing conflicting theories

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we are not doing personal battle

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I am asking that we look at the ideas behind math

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and recognize that they don't fit

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only a novice would think that their studies were truth at first sight

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I seldom use an equation but I am quite well verse in mathematical ideas

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please, don't try to bullshit me with equations

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Someone comes in and posts

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$\bR ^n$

cerulean atlasBOT
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rockhoven

699627666319605861.png
elder gust
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and everyone stops dead in their tracks?

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these equations are absolutely meaningless without context

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Einstein prepares us with a ton of context when introducing E=mC2

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He talks about ideas

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Now we have someone who want to define a point as

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$\bR ^n$

cerulean atlasBOT
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rockhoven

699627666319605861.png
elder gust
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without any reference to the historical literature on the subject?

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we still have not gotten anyone to cite any historical literature on any mathematical idea under debate

quiet lake
quiet lake
quiet lake
quiet lake
quiet lake
quiet lake
#

$\bR$ is the set of all real numbers

cerulean atlasBOT
#

Djake3tooth

616165347552395271.png
quiet lake
#

$\bR^n$ is the set of all n-tuples of reals. Because i wrote n it is by default a natural number (0,1,2,...)

cerulean atlasBOT
#

Djake3tooth

616165347552395271.png
quiet lake
cerulean atlasBOT
#

Djake3tooth

616165347552395271.png
quiet lake
quiet lake
quiet lake
quiet lake
cerulean atlasBOT
#

Djake3tooth

616165347552395271.png
quiet lake
quiet lake
quiet lake
quiet lake
quiet lake
quiet lake
elder gust
#

wdym

quiet lake
elder gust
#

all of them

stone cipher
quiet lake
# elder gust all of them

You asked to define a triangle. I say 'define what?'.
Why are you dwelling on history?
The title of this discussion is "Define a Triangle".
If you accept modern maths there already is a definition of a Triangle, albeit context dependent. So you can't define it again. Only redefine.
If you have another definition on your mind then, please, tell us because idk what you're rambling on about.
If you somehow purposefully forgot the definition of a Triangle, then the question makes no sense. It would be like "Define a jpxfrt".
๏ปฟ
I think you are critiquing modern maths and arguing that Euclid's definition is better.

idle basin
#

why are people tryharding to redefine something that is already fixed

#

its like saying "I'm christian, but i don't like being a christian, so i'm going to make my own religion"

#

@elder gust it says there you're also a middleschool student

#

you should really be trying to do these kinds of stuff until you're like an undergggraduate

#

our minds are too small to comprehend things that you try to do

#

if youre trying to do this then why dont you try discuss the definition of life as well

#

i doubt you'll get far

#
  • you've been yapping a lot
#

instead of trying to "redefine" a shape or asking stupid questions like "why" or "what is life" you should just help people do their assignments at #1020426321261756536 since youre sooooooo advanced and genius

#

good luck, sir yaps a lot,

#

you'll need it

lunar jewel
#

A triangle is one that encompasses the thought of a singular chip of doritos.

elder gust
#

Neither are there any squares or circles or any other geometrical objects

#

They do not exist

#

Points and lines have no existence

#

It is not possible to draw any geometrical object as it is defined

#

Maybe Euclid's triangle is an exception?

#

But that is because his definition is so bare-bones

#

Is a line defined?

#

I think it is defined as length without breadth

#

That is not possible to draw

#

Since triangles are made of lines, it is not possible to draw a triangle

#

Neither is it possible to draw a point

#

Since a point has no dimension

#

and is defined as that which has no parts

idle basin
#

7/10 ragebait try harder loser

lunar jewel
#

@idle basin stop being toxic

idle basin
#

i genuinely think its a ragebait

#

because

#

why would something that already exists have to re-exist

#

why would somethingg that is already well defined have to be re-defined

#

the question rockhoven asked was "define a triangle"

#

you guys defined it

#

but he declines your answer

#

not trying to be toxic or whatever

#

but discussions like there are equivalent to

#

"what's 1+1?"

#

when others answer "it's 2" people still try to ask "but why?"

#

it doesn't make sense that you're denying a fixed definition of a number that already exists since the bronze age

quiet lake
#

Still haven't seen common ground we can work with

idle basin
#

agree

sand ivy
#

Image1

sin(x) = AB/AC
cos(x) = BC/AC
tan(x) = AB/BC
AB^2/AC^2 + BC^2/AC^2 = 1
(AB^2 + BC^2)/AC^2 = 1
(sqrt(AB^2) + sqrt(BC^2))/sqrt(AC^2) = sqrt(1) = 1
(|AB| + |BC|) / |AC| = 1
|AB| + |BC| = 1 * |AC| = |AC|

  • In all right triangles a side is always complex

Image 2

As angles are 90, 45, 45
ratio 1 : 1 : sqrt(2)

ABC = DBC
FC + FD + DC = s1
AF + FD + AC = s2
FC + FB + BC = s3
AF + FB + AB = s4

AD = x
AC^2 = x^2 + x^2 = 2x^2
sqrt(AC^2) = xsqrt(2)
|AC| = xsqrt(2)

Does this make sense @elder gust

second image has 45, 45, 90 traingle
first have 90, x, 90-x

Screenshot_2024-06-22_192342.png Screenshot_2024-06-22_193216.png
elder gust
#

test

elder gust
#

I think it can be shown that they can have one common point anyway

#

Take a look at Descartes premise for The Geometry

#

I see that you are equating rational with congruent

#

This makes sense

#

we can also use the term commensurate

#

You are using two lines to indicate rational?

#

and using three lines to indicate irrational?

#

what is it they you want to prove or demonstrate?

sand ivy
#

Limitation of length of the triangle with the sides

elder gust
sand ivy
#

Idk

#

@elder gust

elder gust
#

OK

#

maybe I should periodically update the first post so people know where we are

#

However, there is no Jump to Top function here and there should be

sand ivy
#

Go on

elder gust
#

How would you begin to define a triangle? Search the definition of a triangle in the historical literature of mathematics. How does Euclid define a triangle? Did anyone else before or after Euclid define a triangle?

#

What do the other posters in this topic offer for definitions of a triangle?

#

Do you agree with their definitions?

#

You might want to begin with Euclid and branch out from there.

elder gust
#

Then draw a non dimensional point

#

According to Euclid, a line is length without breadth

#

Draw me such a line

#

Here is a what we call a point . It has dimension therefore it does not meet the definition of a point being non dimensional. Therefore, it is not a point, in truth.

#

A line is supposedly made of a series of points with gaps between them

#

therefore there is no certainty that two lines intersect at a common point

#

because they just as well could intersect at their common gaps

#

then an irrational line could intersect with two other lines without having two common points

#

if the unit in a line consists of both the point and it adjacent gap

#

what could follow from that?

#

we would end up filling the gap with points

#

and dividing the point

#

this would prove that the original point was not a point at all

#

because according to Euclid a point is that which has no part

#

however we have already divided the point by defining it as having two parts

#

the point and it's adjacent gap

sand ivy
elder gust
#

interesting

sand ivy
#

That meeting point is it

elder gust
#

but that is merely a convention

sand ivy
#

It has a position

elder gust
#

we are saying that there is a point there

sand ivy
#

X, Y

elder gust
#

ok

#

by definition

#

but what are

sand ivy
elder gust
#

we defining?

#

the point or the intersection?

sand ivy
elder gust
#

i see what you mean

#

and I take your POINT

sand ivy
elder gust
#

Do you understand my POINT?

#

The problem is with the unit

#

it is not with either you or myself

sand ivy
quiet lake
sand ivy
#

If line is just 1 dimensional

#

The point will be one dimensional

elder gust
#

I think we unknowingly divided the point when we introduced zero into the number line

quiet lake
#

how so?

elder gust
#

if the line is two dimensional

#

the point will be one dimensional

#

that's a good way to turn this also

#

it is relative

#

but I don't know where that definition of the point comes from

quiet lake
#

maybe projective geometry

elder gust
#

Euclid defines a point as that which has no part

quiet lake
elder gust
#

It is also defined in modern times as non dimensional

sand ivy
elder gust
#

Euclid's definitions suit his theory

sand ivy
#

The point is dependent on the lines

elder gust
#

a unit can be expessed as have any dimension you choose

quiet lake
elder gust
#

1 can be a point, line, circle, triangle, square or higher dimensions

sand ivy
elder gust
#

but if we see that then we should also acknowledge that arithmetic and geometry parallel each other exactly

#

well

#

wait

sand ivy
#

I think

#

Circle too

elder gust
#

Euclid defines a line as one dimensional since it is length without breadth

#

but if a line is one dimensional it follows that it's points are non dimensional

#

I think it wise to first consult the historical literature

#

I did not make up any of these definitions

#

they are either in the historical lit or they are in common circulation

#

what do you know about the definition of a point?

#

Is it unheard of that a point is non dimensional?

#

That is not my definition

#

this too depends

#

if we say that a cube is 3D

#

then a square must be 2D

#

and a line 1D

#

and then what is a point?

#

non-D?

#

How would you figure it?

sand ivy
elder gust
#

in any case, I say that it is impossible to draw these figures and they have no counterparts in reality

#

for example, there are no spheres

#

there are objects that are sphere-like

#

they are spherical

#

but no planet is a perfect sphere and there are no natural perfect spheres

sand ivy
sand ivy
elder gust
#

likewise there are no straight lines

#

they are sphere-like

sand ivy
elder gust
#

they tend toward an imaginery ideal

#

for whatever reasons or causes

sand ivy
elder gust
#

there are no such objects and it is impossible to draw them

sand ivy
#

Its not conspiracy

elder gust
#

where are there actual perfect spheres in nature?

#

or made by humans?

#

you cannot make a machine that will produce a perfect sphere

#

you are referring to you senses

#

which are incapable of noting the imperfections

#

like when astronomers first began looking at the sky

#

they thought the planets were perfect spheres because they viewed them with the naked eye

#

and at a great distance

sand ivy
elder gust
#

you cannot check enough

sand ivy
elder gust
#

don't attempt to defend the impossible

#

you can not draw a straight line

#

or a right angle

#

it is impossible

#

you are just looking at it and refusing to acknowledge it's imperfections

#

we call it a point

#

. this point can be divided into pixels

#

same with this point .

#

...........

#

all of these points have parts

#

and therefore they are not points according to definition

#

but select a definition and follow through on it's consequences

#

what do you think of the other definitions that posters have provided us with?

#

there have been plenty of thoughtful comments

#

engage those posters

#

and search out the truth of the matter

#

i have little interest in engagement with those who simply want to prop up conventions

#

I am in search of truth

#

meaning at least that the things that are propounded are consistent with themselves or other known thro

#

theories from the historical record

sand ivy
quiet lake
quiet lake
sand ivy
elder gust
#

It's your turn

quiet lake
quiet lake
sand ivy
elder gust
elder gust
elder gust
elder gust
#

This is problematic

#

I a very real way, zero complicates the idea of dimension

#

If a point is the basic building block of geometry

#

and a point is analogous with the number 1

#

then we have more problems

elder gust
#

the diagram you provided needs only some development for this end

ionic laurel
#

You will not get anything so perfect irl

quiet lake
elder gust
# ionic laurel Same with rational lines

well, there are no such things as rational lines. We designate a line as rational and it serves as a unit to measure any other line. Euclid has two or three books on lines.

#

...................

#

that's a line made of points

#

we can designate this line as rational

#

then measure other lines with it

#

and all of this is pertinent to the conversation

#

let's take any two lines

#

they can have a common point simply by intersecting them

#

well the simplest cases

#

a line of two ..

#

and of three ...

#

they are prime to each other

#

2 ..

#

4 ....

#

2 can measure 4

#

anyway

#

any two lines rational or irrational can intersect at a common point

#

How can we determine whether an irrational line can have two common points with

#

two other lines in a right triangle

#

we have some congruencies marked in this diagram

#

which lines would you want to designate as rational?

#

we should do the problem both ways for best illumination of it

#

what we need first is to understand the problem

#

that is as far as we can go

#

because i dont believe there is any solution

ionic laurel
elder gust
#

but just to understand the problem

elder gust
#

that is why math is so full of redundancies, equivocations, and ambiguities

#

from the unit all of the way to infinity

#

and math has evolved just like english or any language

#

so modern math is like modern english

#

but the middlle period up to the Enlightenment is another language

#

it is like middle english

#

we can understand it but it requires a little bit of work

#

***Whan that Aprille with his shoures soote,
The droghte of March hath perced to the roote,
And bathed every veyne in swich licรณur
Of which vertรบ engendred is the flour;
Whan Zephirus eek with his swete breeth
Inspired hath in every holt and heeth
The tendre croppes, and the yonge sonne
Hath in the Ram his halfe cours y-ronne,
And smale foweles maken melodye,
That slepen al the nyght with open ye,
So priketh hem Natรบre in hir corages,
Thanne longen folk to goon on pilgrimages, ***

#

you can decipher this easily with a little effort

#

but ancient math reads like old english

#

even Viete is very obscure and that is not as old as Euclid, Nicomachus, Arhimedes or Apollonius

#

that was the Prologue to the Canterbury Tales

#

Wait and I'll dig up some old english

#

Hwรฆt. We Gardena in geardagum,
รพeodcyninga, รพrym gefrunon,
hu รฐa รฆรพelingas ellen fremedon.
Oft Scyld Scefing sceaรพena รพreatum,
monegum mรฆgรพum, meodosetla ofteah,
egsode eorlas. Syรฐรฐan รฆrest wearรฐ
feasceaft funden, he รพรฆs frofre gebad,
weox under wolcnum, weorรฐmyndum รพah,
oรฐรพรฆt him รฆghwylc รพara ymbsittendra
ofer hronrade hyran scolde,
gomban gyldan. รพรฆt wรฆs god cyning.

#

That is Beowulf in old english

#

Since math is a language it is prone to the same circumstances

#

there are many accidents that have occurred in it's development

#

languages don't evolve all nice and neatly

#

a word that was coined 2500 years ago may mean the exact opposite in today's english

#

that is what we have with the unit and the point

#

very strange twists of meanings

#

they are entirely different objects of thought today from how they were first conceived

#

because they spring from human imagination

#

and math is a conglomeration of a multitude of imaginations

#

coming from different populations and individuals

#

people do not count uniformly from culture to culture

#

even in the present day

#

for example

#

I could say let's meet at 1:00

#

and that means on the POINT

#

1 is a point

#

but not in some cultures

#

in some places we could agree to meet at 1:00 and the person could arrive at 1:45 and not be late

#

that is because they count the point and the adjacent space

#

they count the 1st hour

#

1:00 means any time within a limit

#

and the unit has taken on these meaning throughout the history of math

#

without notification

#

so I say that the number one was accidentally divided

#

the conditions for this accident were introduced with the admission of 0 to the number line

#

as long as we began the number line with 1 there could be no division of 1

#

and there was none

#

when we introduced 0 on the number line, we admitted a space before the number 1

#

and it is that space which admits of division

elder gust
#

Now, thinking about

#

calling the point non dimensional is a problem

#

because I find an analogy between the point and the number 1

#

but if the point is non dimensional it should be analogous to 0

#

so you see how much confusion is actually latent within math

#

due to the various interpretations that have been injected into it

#

and we can trace this back to the earliest records

#

here are what may have been some of the earliest numbers

#

they are just shapes

#

but not that the units are counted as single discrete points

#

we don't count the spaces

#

then look at any modern ruler

#

look at a carpenter's tape measure

#

it starts with 0

#

that is a whole other animal

#

it's not at all the same way of counting

#

in the shapes theory, the count begins with 1

#

and on a modern ruler the count begins with 0 a space and then the point that marks 1

#

does 0 deserve a point on the number line?

#

because IDK

#

in coordinate geometry sometimes 0 is marked with a point

#

isn't it more common to mark it with the letter O

#

which means origin

#

but that admits more confusion

#

because someone recently said that 0 was the origin of number

#

as in the successor theory

#

right

#

but in that theory 1 is the origin of other numbers

elder gust
#

What do you mean by reals and vector spaces?

#

#1254508004636758218 message

#

Use this link to get to the top of any of our discussions and review the material.

quiet lake
#

I'm gonna start with the latter. A vector space is one way to build up space given a line (the ground field).

#

Reference: https://en.m.wikipedia.org/wiki/Vector_space

Vector space

In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called vectors, may be added together and multiplied ("scaled") by numbers called scalars. Scalars are often real numbers, but can be complex numbers or, more generally, elements of any field. The operations of vector addition and scalar multi...

#

You can implement them by putting numbers in tupels, like (x,y,z)

#

Next, the reals. These are harder to formalize.

#

Intuitively they are infinite-precision numbers. They have an unending amount of decimals.

quiet lake
#

More formally you can do this by looking at sets of rationals, e.g. sqrt(2)={rationals q with q<0 or q^2=2}, or by making sequences of rationals, e.g. (a_n) with a_(n+1) = (a_n)/2 + 1/(a_n).

#

Reference:
https://en.m.wikipedia.org/wiki/Real_number

Real number

In mathematics, a real number is a number that can be used to measure a continuous one-dimensional quantity such as a distance, duration or temperature. Here, continuous means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion.
The real numbers are fu...

sand ivy
elder gust
#

What do you mean by the word triangle? That is the question.

sand ivy
elder gust
#

Where did you get this definition?

#

I think we could clean this up a bit

#

Have you reviewed the thread? What do you think of the other definitions supplied?

#

You should be able to get to the top of this discussion through the selection of links

#

#1254508004636758218 message

#

If I post the link directly here it will bring you to the bottom of this thread.

#

But if you bookmark the Links for review thread you can always get back to the top of the discussion.

sand ivy
elder gust
#

this is your own personal interpretation?

#

how do you know that what you were looking at was a triangle?

#

who told you?

#

i remember the teacher pointing to a tree sides object and saying "triangle"

#

do you subscribe to any one else's definition?

#

Do you know how the master mathematicians have defined a triangle

#

and do you agree with them or disagree with them?

#

Zeta: A 3 sided polygon were a point of a line segment is joined with another line with point were both lines don't touch vertically and another line connected to points of the unconnected in both which is not touching vertically or horizontally

#

why do you need segments in your definition?

#

What do you mean by joined?

#

Are you indicating a vertex?

#

and a closed space?

sand ivy
elder gust
#

If it is a right triangle how could we determine whether the irrational diagonal could "join"

#

with another line at a point?

sand ivy
#

<--------->

elder gust
#

how do you make a triangle?

sand ivy
elder gust
#

What is spose to be "joined" in your triangle?

#

How are these lines to be "connected"?

sand ivy
elder gust
#

Where is the end point of the irrational diagonal?

sand ivy
#

โˆš(x2 - x1)ยฒ + (y2 - y1)ยฒ = length of irrational
โˆš((x2 - 0)ยฒ + (y2 - 0)ยฒ) = length of irrational
Length of irrational is โˆšAB
Solve

#

@elder gust

elder gust
#

where is the end point of the irrational diagonal?

#

can you find the endpoint of pi?

#

if you could find the end point of an irrational line

#

you could find the end point of an irrational number

#

we don't yet have anything to solve

#

we are merely exposing the issue

#

we are outlining an unsolvable philosophical problem

#

and exploring the dilemmas

elder gust
#

@sand ivy

sand ivy
#

If you can't then i will

quiet lake
#

this is only possible if you can get the number using rationals, adding, multiplying and square roots

#

thus you can't construct a line of length pi from a line of length 1

elder gust
#

I think we have some work to do

#

since the sqrt2 is irrational i can see no way possible for it to "join" or "connect"

#

with a common point

#

But I think that this can be demonstrated by just measuring the lines with a common unit

#

only one end of the irrational line can join with a common point

#

the other end will miss that point

#

if it we make it to join with a common point

#

this will result in a misalignment of the "right" angle

#

the way I see it is you can not have both situations

#

you can either have a connection at both common points with the irrational

#

or you can have the right angle

#

so I take exception to any reference to a "right triangle"

#

If there is a set of three connections in this isosceles triangle

#

it can may also be because the nature of the point is redefined at one connection

#

it is not the same kind of connection at each of the vertexes

#

on two vertexes there is a true superposition of points

#

but on one connection it is not the points that connect

#

but the intersection is made in the spaces between points

#

and that connection can only be expressed in terms of a limit

#

we cannot identify any fixed loci

#

in an irrational line or number one end can be fixed to a point

#

but at the other end the point will always escape exact measurement

quiet lake
#

oh wait

#

right

elder gust
#

if lines are composed of discrete units separated by spaces which indicates their discreteness

quiet lake
#

not isosceles

elder gust
#

such as ..................

#

well it has two equal sides

#

though you might be taking Hilbert's argument?

#

but a line .......................

#

why can't two of these lines intersect at a common space

quiet lake
elder gust
#

rather tha a point?

#

I have no idea

#

There may be many definitions

#

and it depends on how you view the point

#

just as number theory depends upon how you view the number 1

quiet lake
#

yes but which definitions are you using right now

elder gust
#

why is it that when two lines intersect we are justified in marking it with a point?

#

I am not sure which definitions there are or which I am using

#

good question

#

what do you think?

#

I mean for communication

#

we have to consider both sides

#

what do you think when I say "point" and "line"?

#

all that matters is that we keep track of what symbols we are using

quiet lake
#

i think you're defining lines as points interspaced regularly

elder gust
#

and what meanings are attached to them

elder gust
#

Would you agree to this definition?

#

That is what matters

#

and what other options are there?

#

as long as we keep track of everything

#

there should be no argument

#

we should not waste effort on avoidable equivocations

quiet lake
#

but it's an interesting idea