help-23

[glacial pine]

i got 11, im not sure what i did wrong

[fathom adder]

24/3 = 8 not 6

[glacial pine]

oh my god.

[safe radish]

@glacial pine Has your question been resolved?

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[left gyro]

.@tulip plover no thats not correct, you find the h by subtracting √3 repeatedly, just using 48 and 24 is not going to cut it
use 48 - 14√3
(also I just woke up)

[tulip plover]

the issue is how do I figure out how much do I have to position it vertically

[tulip plover]

the smaller triangle should always have equal gaps from the bigger one as long as it's in the middle?

[left gyro]

thats not true

[tulip plover]

so then I need to figure

[left gyro]

oh btw did you sleep

[tulip plover]

which smaller ones aren't available

yes

[left gyro]

great, because youre asking questions we already answered

[left gyro]

tell me again, what do we do do h to move this 1 unit to the right?

[tulip plover]

decrease it

[left gyro]

ok lets be more specific on this

[tulip plover]

by sqrt 3

[left gyro]

that is not at all a helpful answer

there you go

moving it in general to the right requires that you decrease h

[tulip plover]

yes and if we do that 14 times

it will move 14 blocks inward

but you said that doesn't make the gaps equal

[left gyro]

I didnt say that

[tulip plover]

I asked that?

[left gyro]

you did 24 instead of 48 - 14√3

[tulip plover]

no I didn't

I was talking about what you were talking

[left gyro]

then in that case, the answer is also no

you need to account for the fact that the resulting triangle that's generated is not centered vertically

wait

[tulip plover]

that's why I was assuming that it is in the center when asking

[left gyro]

oh that's how we can do this

[left gyro]

there's a way we can go and do this

alr this is (w, h) = (0, 34.5)

now say we want a concentric triangle that is close to half the size, with exactly equal gaps

hold on Ill get a proper example of the triangle first

@tulip plover here we go

hopefully its evident from the picture what I did to create the triangle

[tulip plover]

the same thing I've been saying?

[left gyro]

tell me

how many times did I subtract √3?

[tulip plover]

that is not what I was asking about

[left gyro]

if you dont know the number of times

[tulip plover]

you did it 7 times

what I am asking about

is we know that h - x√3 will decrease it by one and make it smaller but what I am asking is how do we know that the smaller triangle is going to have equal gaps between it and the bigger one because as you said it's not always going to be equal gaps so we need to isolate all the ones that are equal and exclude the ones that aren't equal

[left gyro]

I am talking in a general sense

[tulip plover]

so then tell for this tool specifically

[left gyro]

you made this statement which is not true for all subtractions

for this tool specifically, the statement is still not true beause subtracting √3 is not the only thing you can do

you need to specifically talk about subtracting √3 for the statement to be true

and there the gaps are indeed equal

[left gyro]

as for the other "no", I made a mistake

[tulip plover]

for every √3 h is decreased by from the bigger triangle the smaller triangle will always have equal gaps with the bigger triangle

[left gyro]

I thought 48 - 14√3 was correct, but its not

it would have equal gaps but it wouldnt be of the correct size

[tulip plover]

hmm

what do you mean not the correct size?

distortion as it gets smaller?

[left gyro]

because although its close to h/2,

the block gaps only align with the bottom half of the triangle

but we need them to align with the middle half instead

so we need to pick a different size, then cut it off at the bottom, for the blocks to align properly

simply making an h/2 triangle by subtracting √3 from h, then moving it upwards, wont work

[tulip plover]

so a bigger triangle that starts at the base of the bigger triangle?

[left gyro]

a bigger triangle where (w, h) is at the tip of the smaller triangle's position

my calculator is lagging, Im fixing that rn but Ill show a better example

to generate this triangle here,

first, the smaller triangle in the center has (w, 2/3 h)

but I cut it off at the bottom so it doesnt go all the way

the purple triangle is really a triangle with (w, 1/2 h) shifted upwards by 1/6 h

[left gyro]

knowing the smaller triangle is 2/3 h, we approximate h - n√3 ≈ 2/3 h

which leads to n ≈ h / (3√3)

this is why 7 works for 34.5

now there is something sneaky we can do here

remember how we said earlier that rational approximations of √3 were too large to be usable?

[tulip plover]

are you going to scale them down?

[left gyro]

not exactly

Im going to use a multiple of √3 for the height

[left gyro]

that way the n here is exact

and if we pick the correct multiple, we can see that h is not far off from an integer height

so it looks alright

[tulip plover]

I see

[left gyro]

if there arent many terms in the continued fraction,

the denominators are more usable

but we dont need to rely on these actually

you could just keep testing various multiples of √3 until its close to an integer

[left gyro]

I dont know the proof, but these continued fraction ones are the best you can get

[tulip plover]

so trial and error is unavoidable?

[left gyro]

if you want to avoid trial and error, use a continued fraction

[left gyro]

this says √3 ≈ 26/15

so 15√3 ≈ 26

do you know how to make this continued fraction manually?

wait nvm you dont need to know that

you could just start from here and erase terms since 97/56 is too large

that means 7/4 and 26/15

so really just 26/15 since 7/4 is too small

[tulip plover]

isn't it just multiplying the numerator and denominator with the same value?

[left gyro]

not really

here are the first steps

you start with the number
then you add/subtract it until it is at its closest to 0
then you 1/
then you repeat

because √3 is whats known as "quadratic" (its a solution to a quadratic equation), the continued fraction eventually takes on a pattern

so repeated steps will do this

regardless, when you want to stop the method, you can see in the above that this is supposed to = 0

put = 0 at the end, then solve for √3

[tulip plover]

I see

[left gyro]

here's the end result

you can see theres a pattern here so you dont have to do this the next time around

this pattern happens for square roots

due to our process, you can imagine that this is in fact a sequence that can converge to √3

if you solve for the limit, you get ±√3

[tulip plover]

yes I see

[left gyro]

so this is definitely a sequence that converges to √3, and (unprovenly) also gives the best rational approximations at each step

[tulip plover]

so x approaching sqrt 3?

[left gyro]

yes

now keep in mind these rational approximations can estimate 15√3 to be an integer

but you could also just try out various heights

the only think you need to look for is at the bottom, whether the outer triangle appears acceptable or not

if visually you cant tell the difference, you might as well just use any multiple of √3 and not need to resort to 15√3 only

here's the resulting triangles that appear

except for the base, Id say there is barely any error involved here

because h = 15 √3, that means h/(3√3) = 5

so h - 5√3 = 2/3 h

but since h ≈ 26, that also means 2/3 h ≈ 17

so you both get an exact rendition of 2/3 h and you also see that the error for h itself is extremely small

well 2/3 h ≈ 2/3 * 16 = 17 1/3 though, so the error isnt as good as it could be

but thats a lot of spots to look for error, let me see

[tulip plover]

okay I see

[left gyro]

now 15 √3 isnt the only number that works

[tulip plover]

so if you want half the triangle you would do 1/2h which is 13

[left gyro]

wdym

[left gyro]

this is doing half the triangle

we do that with 2/3 h instead of 1/2 h

[tulip plover]

okay so we are actually forming a triangle that is 2/3 the size of the bigger triangle

but we cut it at the top to get a triangle that is half the size

[left gyro]

we cut it at the bottom

you can see that the bottom, the lines continue towards the base of the bigger triangle

[tulip plover]

yes

[left gyro]

if you just not build those and draw a horizontal straight line where the base of the smaller triangle should be, that would be it

[tulip plover]

but the actual scale of that triangle is 2/3 the size of the bigger triangle

[left gyro]

the scale of the triangle is 1/2 the size of the bigger triangle

you dont seem to be remembering my words every well

[tulip plover]

you mean the purple one is half

[left gyro]

yes

[tulip plover]

I am talking about the triangle we made to form the purple one

[left gyro]

then dont use the word "actual"

its ambiguous

[tulip plover]

that is the actual triangle?

we cut it off

to make the smaller one

[left gyro]

actual can be either in the sense of "its real" or "its ideal"

"its real" is 2/3

"its ideal" is 1/2

[tulip plover]

hmm

[left gyro]

dont use the word

[tulip plover]

okay

[left gyro]

use practical, or empirical

[tulip plover]

how do you know where to cut off the bottom

oh I guess since the gaps are equal

the bottom gap would be the same as the side gaps

[left gyro]

@tulip plover back

anything else?

now that I think about, thats both parts of the question answered

the tetrahedron and the inner concentric equal gap triangle

[tulip plover]

30 sqrt 3

and 20 sqrt 3

[left gyro]

if you insist

oh right

[left gyro]

I need to give you a more convenient version

[safe radish]

@tulip plover Has your question been resolved?

[safe radish]

@tulip plover Has your question been resolved?

[left gyro]

@tulip plover polished version https://www.desmos.com/calculator/razdkebern

renamed (w, h) to (x0, y0)

d controls the proportion of the inner blue triangle (0% to 100% size)

erase/place a comma here to choose whether to draw the triangle with equal gaps or not

it should be a lot easier to for example trial and error various equal gapped triangles just by moving the sliders around

you can move the point anywhere

if one of the directions is locked then replace y0 with a decimal number

[tulip plover]

I see

I have a question about this point here

so the base uses 0 only when x = .5

this seems irrelevant to the height of the triangle

no matter what value you plug into it will always have the same corner

and it looks different from the inner triangle

I think this is unavoidable due to the equal gaps of the sides

oh I see why the bigger triangle has those missing corners

nevermind

I thought the corners only appeared in odd integers

but it seems no value you plug into x sqrt 3 leads to that corner appearing

that's fine regardless

[safe radish]

@tulip plover Has your question been resolved?

[left gyro]

@tulip plover what are you left to do rn

[tulip plover]

I was asking the question

also is this the finished product basically?

[left gyro]

yes

the link answers the question

please pay closer attention for the answer was already told to you

[tulip plover]

hmm

[left gyro]

we cant position it vertically

the best we can do is choose a smaller h by subtracting √3

this will ensure equal gaps

[tulip plover]

not that question

you answered that

[tulip plover]

this one

[left gyro]

oh right this

very specific circumstances so I wouldnt pay too close attention

as a reminder, the finished triangle wouldnt readily show you what the original red and blue triangles are

[tulip plover]

hmm but the inner triangle doesn't have the same looking corner of the big one

[left gyro]

oh thats no good

Im seeing that its good enough to slightly tweak d

it also helps to use the side of the triangle as a guide

if for example you want these corners

there needs to be a 2x1 block

[left gyro]

unfortunately I dont have a better answer than that

already its rather restrictive to have equal gaps in the first place

the d slider is about the most control you can have

[tulip plover]

it is because of the 1 block length sides

[left gyro]

yep

and every time we - √3 we advance by a block

I think the d slider is the best way of going through them

it remains proportional if you increase/decrease y0

[tulip plover]

well it's fine as long as it's the most accurate my current big triangle already has a similar shape

[left gyro]

alr

oh theres a simple way I can actually fix a particular problem here

[tulip plover]

hmm

[left gyro]

desmos has a new feature where when I save, you see an updated copy

just refresh the page and you might see a minor difference here:

Ive changed the blocks on the bottommost row of the inner triangle so that the gaps remain the same there as well

keep in mind you could just not place the blocks there if you dont want to place them

[tulip plover]

yes

I think it is satisfactory I will try to figure out how to make the inner one into a tetrahedron and see

[left gyro]

try using the 3D link but with a non-integer height

the side length of the blue triangle is written as s1

you can see line 19 for that

or just type s1 on a separate line

you can change the slider to not snap to integers by tapping the bounds

nvm I just removed it, thats easier

[tulip plover]

so the previous line 19 is removed?

[left gyro]

oh my bad

[left gyro]

this blue triangle is in the 2D link

the inner triangle is the blue triangle in the 2D link

so line 19 is of the 2D link, you can find line 19 there or alternatively you can type s1 on a separate line to read it

[tulip plover]

and I plug that in here?

[left gyro]

yep, type it in

Ill need to check

but this s was originally intended to count the number of blocks (fencepost 1)

theres a good chance the tetrahedron you get wont have the same base as the triangle

currently the 3D version doesnt have a way around this, Ill need to rewrite it to be (x0, y0, z0) for that to be fixed

Im not intent on doing that rn

[tulip plover]

it seems to work fine with non integer heights

[left gyro]

but the base may not exactly align with the innter triangle in the other link

[tulip plover]

hmm

[left gyro]

also, there seems to be like a 2px gap on the bottom of your screen

I saw it a few times, idk whats up with that

[tulip plover]

I use hyprland

it's a linux thing

window tiling

[left gyro]

I see

[left gyro]

the best option I have for you about having this line up isnt very neat

choose y and d so that s is close to an integer

and that the base of the blue triangle is close to a half-integer

that way it should be close enough to the assumptions the 3D link uses

[tulip plover]

yeah that's what I was expecting

[left gyro]

very unfortunate that youve had to do this

you can use this as sort of a reference

wait wrong one

this is essentially what s1 is

[tulip plover]

is there not a way to set a value for s in the 2d one that the other values would change to match it?

[left gyro]

not really

[tulip plover]

is ovee

[left gyro]

desmos isnt exactly built to do that

just choose some values for d and y0 to solve for your s

[tulip plover]

alright

[left gyro]

also, this is the y-coordinate of the base of the blue triangle

for this to be close to a half integer, that means (y0 - y1)/3 needs to be close to an integer

using this,

oh wait nvm thats slightly off, its not read like that

[tulip plover]

is it possible that you could make a diagram of the inner workings of these 2 desmos projects that describe how the lines work in the system and their effect on the triangle

[left gyro]

brb

[tulip plover]

is ovee

[left gyro]

@tulip plover updated the lines to have slightly simpler behavior, I dont remember the changes

[tulip plover]

hmm

y is max 28.75 now?

[left gyro]

as I said before, you can just move the point above 28.75 to change it

[tulip plover]

I know

I was just curious why you made that change

[left gyro]

it doesnt mean anything

[tulip plover]

okay

[left gyro]

it just happened to have changed while I wasnt looking

[tulip plover]

it looks perfect I think?

is this supposed to be the case?

[left gyro]

hold on I thought I reverted that

let me see

oh right yes that intentional behavior

[tulip plover]

okay

[left gyro]

your y1 is set to line 6, not line 5

so the blocks arent fitting the blue triangle exactly

theyre fitting the closest blue triangle that would have equal gaps

[tulip plover]

I see

[left gyro]

the outer red triangle has a base that is 0.5 units above the x-axis and the top at (x0, y0)
its side length is s0 = (2 y0 + 1)/√3
line 24 shows the coordinates of the outer red triangle

0 < d < 1 controls what proportion the inner blue triangle is at
or that the inner blue triangle has side length s1 = d * s0
line 25 shows the coordinates of the inner blue triangle

note that the top coordinate is (x0, ((1 + 2d) y0 + 1 - d)/3)
set y1 to line 5 to have the inner blocks fit the inner blue triangle normally i.e. y1 = (1 + 2d) y0 + 1 - d)/3)
set y1 to line 6 to have the inner blocks instead fit the closest inner triangle that has equal gaps

I dont see anything off in this picture

[tulip plover]

no I'm just showing how it's the same triangle with a .03 difference

and it fits in

[left gyro]

oh alr

[tulip plover]

is this basically perfect for the tetrahedron?

[left gyro]

that looks pretty good

[tulip plover]

or actually I don't know anymore

okay so I changed d to .47 as I showed

and it makes the exactly same triangle as .5

[tulip plover]

that is the result for d = .47

this is the result for d = .5

it's a huge difference despite being the same triangle being formed in the middle

wouldn't there be a d that is the most accurate for s that makes this triangle since it is shown that multiple values can make the same thing

[left gyro]

thats not really necessary

[tulip plover]

hmm

[left gyro]

both s1 values are valid

[tulip plover]

but which is more accurate?

[left gyro]

theyre both accurate

[tulip plover]

how?

the triangle doesn't change at all

[left gyro]

remember youre using line 6 instead of line 5

line 6 as a reminder selects the closest blue triangle with equal gaps

[left gyro]

theyre both accurate in that you cant choose a better equal gapped inner triangle than those

[tulip plover]

no the issue is the tetrahedron not the inner triangle

[left gyro]

change it to line 5 if you want to see the d values be different

[left gyro]

Im going to be honest, the solution is rather bleak

[tulip plover]

because we need to plug in the s that is generated from this graph to the 3d one

but these 2 exactly equal triangles are making 2 different values for s

[left gyro]

alr Ive added an extra line to state the side length of the triangle that the blocks are approximating

then again

oh I see a more reasonable way to go about doing this

[tulip plover]

same exact triangle formed but different s values

[left gyro]

hold on a second this is gonna be more polished

[tulip plover]

okay

[left gyro]

you already told me the same thing three times

refresh your page, a temporary solution is already in place

[tulip plover]

then why talk about line 5?

I was looking at line 19

[left gyro]

I misread what you were talking about

[tulip plover]

so I did need to tell you again

[left gyro]

you didnt need to send an image after I already updated the link

[tulip plover]

ovee

you updated after?

[left gyro]

...here

[tulip plover]

ovee

though I don't see how this solves the most accurate value for s?

huh?

equal gaps brokers

[left gyro]

going to fix that too then

hmmm

[tulip plover]

it's oee

[left gyro]

on second thought I have decided not to fix this

the method to place the base of a triangle is different than the sides

if you want it to line up, just dont place the blocks as stated earlier

[tulip plover]

okay

I was confused what you meant don't place the blocks

you just mean don't do the the corner parts

still that leaves the final issue of s value

[left gyro]

@tulip plover more polish now

nearest equal-gap triangle is now green

the original d s0 inner triangle without equal-gap guarantees is blue

also the controls have been updated to (X,Y) for the center

[tulip plover]

I see

does this mean the green one has a more accurate s value?

[left gyro]

s value that is accurate to?

[tulip plover]

for the tetrahedron?

[left gyro]

no

I cant really guarantee any of that

[tulip plover]

so there is no way to get the tetrahedron?

not even a manual way?

[left gyro]

ofc you can manually do it

also

[tulip plover]

yes I want the most accurate

[left gyro]

you know the difference between the blue and the green inner triangle?

[tulip plover]

yes

the green one goes solely for the equal gaps as you said

[left gyro]

there are two different s values for each

[tulip plover]

it has se

[left gyro]

yes, so you can just match it up with se

[tulip plover]

the issue is checking that the value of the s is accurate to the triangle

how do you check it or find the accurate s value

[left gyro]

as in accurate to the base of the tetrahedron?

[tulip plover]

do you not remember what I was saying about d=.47 and d=.5 making the exact same triangle but the value of s was 16 and 17

[left gyro]

and youre saying se doesnt fix this?

the purpose of se is to tell you the side length of the green triangle

[tulip plover]

I am asking you if you did it to fix that

[left gyro]

yes?

[tulip plover]

oh okay

[left gyro]

Im not sure exactly what you were asking

if you drag the centroid around,

you can see the green triangle is less sensitive

its as equally insensitive as the equal-gap blocks

you can think that its the triangle that those equal-gap blocks are approximating

[tulip plover]

yes I just tested it

the green triangle does solve the issue

[left gyro]

theres also having it be separate triangles

[tulip plover]

the s value only changes when the grid triangle changes

[left gyro]

blue and green

we are reaching previously unforetold levels of polish

[tulip plover]

2 different d values and same s value when they make the exact triangle

[left gyro]

very good

[tulip plover]

I think this is officially complete it should be able to form the tetrahedron if the s value is accurate

[left gyro]

if anything, the next order of business is to generalize the 3D link

[tulip plover]

hmm

[left gyro]

but I dont have the heart for that

[tulip plover]

is ovee

[left gyro]

lot of tedious work

[tulip plover]

I'd like to know the manual way to make the tetrahedron

I think that will yield the best results?

[left gyro]

we know the tetrahedron is generated from assuming the corner is at (0.5, 0.5, 0.5)

so it should be enough to make a triangle that has this lower-left corner

if you want exact results, youll need to solve for the necessary X, Y

I dont think its as bad as it looks

for one you can observe that the y-coordinate of the base of the green triangle seems locked into particular places

you wanted the inner triangular prism to be how wide?

[tulip plover]

around 25?

20-28

the size is relative to the bigger triangle

I think this would be the triangle I make

or this one

figuring out how to make the tetrahedron of the inner and outer triangles of these 2 is all I need

I just moved the x to 1

[left gyro]

Im finding (X, Y) = (0, 14.67) looks pretty good

the only conditions are that the the lower-left corner of each triangle is close to the center of the block

with that, you take the s0 and se values and then find the corresponding tetrahedra

as long as the values are exact enough (or you adjust Y slightly if its wrong) you should get the tetrahedra you want

I will point out that a generated tetrahedra may not exactly match the top half of a larger tetrahedra

hopefully you werent assuming thats the case, you know the back of the $1 bill does leave a slight gap between the tetrahedra

[tulip plover]

I have no idea about the dollar bill I just wanted tetrahedron

[left gyro]

Im talking about hiding this mismatch with a gap between the larger and smaller tetrahedra

[tulip plover]

oh

that's fine

I was kind of thinking of doing that

flattening the top

[left gyro]

well no other choice now than to try and do it

Ill see you in a bit

[tulip plover]

I was saying that I didn't want to generate it and wanted to make it manually 😭

guess this counts as close to the center of the block

plugged the bigger triangle's s value and I got this

very hard to count these balls

plugged the smaller triangle in and got this

seems the balls converge with each other in the edges of the faces

I guess I can build this face and try to find the elevation of each ball?

the elevation is visually confusing

yeah this is why I wanted to manually do it this is too visually confusing for me to interpret how I am supposed to place the blocks

[tulip plover]

oh I just saw the z slider

[tulip plover]

this does not work

I got the value of the inner triangle but the first layer of the tetrahedron does not even match it

I think it's because the double tip it alters the slope a little

that's why I think a more manual way is needed for this

[safe radish]

@tulip plover Has your question been resolved?

[safe radish]

@tulip plover Has your question been resolved?

[safe radish]

@tulip plover Has your question been resolved?

[left gyro]

you have to listen to what Im saying next time

I showed you the capital Z slider twice

[tulip plover]

well I figured it out but still didn't work

[left gyro]

oh right, I forgot to update this to handle non-integer values

thats a relatively quick fix

[tulip plover]

and then it is completely finished

[left gyro]

you know if you bothered to actually read the lines, you could figure this out and not have to rely on me to do it all for you

[tulip plover]

this is foreign to me and that is why I was asking you to make a diagram explaining these lines and what they do to the triangle

[left gyro]

oh you mean each line in the desmos, not the drawn lines in the 2D link

well thats not very possible

[tulip plover]

I just want to understand the math

as I said from the start

[left gyro]

youre asking me to understand it for you

also,

[left gyro]

its not possible for each line in the code to have a distinct visual representation in the output

[tulip plover]

no

[left gyro]

so a diagram wouldnt be possible

[tulip plover]

I am asking for just something that shows the math not the output

[left gyro]

the math is there on the left

[tulip plover]

like the workings of the math

[left gyro]

that is also there on the left

it is clearly makred

you will have to read it slowly to understand it

[tulip plover]

I want a very thorough explanation

[left gyro]

listen

you seem to think you know the best on how you want to learn

there are two issues that you have just glossed over while I was talking

[left gyro]

lets start with the first issue

[tulip plover]

I never even heard of floor or ceil before

[left gyro]

please stop interrupting

[left gyro]

elaborate on what a diagram is for me

[tulip plover]

you know on a whiteboard where a teacher draws on a marker labeling the math

and explaining it to you

schematics?

I want the math described

[left gyro]

for a variety of reasons, that is not possible here

let me explain what these reasons are

[tulip plover]

why?

[left gyro]

please stop interrupting

usually for simpler examples, you can read off the lines to see where each part is

however here, each line depends on either a set of variables with special meanings, or a particular way of looking at the triangle with special meanings

a blueprint or schematic does not show this information, it does not show the origin of a design, only its specifics

what you are looking at on the left is a schematic, and as you can see it is not immediately readable

therefore a schematic will not help

a diagram is a more simplified version of the schematic that just consists of a labeled version of the output

as you can imagine this allows less detail than a schematic and so will also not work

to understand the math, you have to understand the approach I used to write that math first, which I have already shown you twice

but I can show it again, it is not clear at first

do you understand?

[tulip plover]

yes

[left gyro]

second reason

[left gyro]

to explain something, the best way to explain something is to do it yourself

this is a hard pill to swallow, but you know best on what you dont know

there are times where this is not the case

for example, youre starting out on something new where you dont know what you dont know, so you can only at best explore a small portion by yourself

it is not the case here, I have already told you the basics of everything required for the 2D version let alone the 3D version

[left gyro]

that includes what floor and ceiling are, I showed them to you and you said you understood

now you are mentioning this like you dont remember, let me repeat what I said:

[tulip plover]

no I do remember

I meant before we speaking

[left gyro]

why refer to the past for a present problem?

[tulip plover]

because it relates to what I am asking for

I am trying to tell you

[left gyro]

hopefully youll say something that I dont know

[tulip plover]

I want just details on how you conceived this so I can understand your approach to this problem so I can learn myself how I can apply similar thinking to other things as from the start I had no real idea how I can tackle this problem and this math you used to form this has not been used by me and I want to know where it stems from and the details that helped you conceptualize a solution for this

[left gyro]

it was already explained in the 2D link

I dont think you understood what I said in the 2D link then

did you lie?

I will explain the 2D link again

this time, be honest when you dont understand something

[tulip plover]

I do understand but I am asking for like a piece of media that summarizes it all

[left gyro]

not everything is made out of 20-minute youtube videos buddy

[tulip plover]

that's why I said diagram

[left gyro]

Im just one guy

[tulip plover]

I am just asking for like a summary 😭

[left gyro]

it doesnt fit in a summary

[tulip plover]

it's so ovee

[left gyro]

it fits in a 20-minute youtube video, but not a summary

you really think that, if it could so easily be explained in a summary, that I wouldnt have just told it to you by now?

this isnt going to fit in 2 sentences bro

lets just repeat what I said from before

this time youre partially familiar with the pieces

[tulip plover]

I was thinking of like a few pictures

[left gyro]

now maybe we can do a summary, but itll require some work from you

[tulip plover]

hmm

[left gyro]

there is exactly one picture, and its the one Im showing you

not all math is visual

let me continue

heres a picture of the triangle, and please dont interrupt

now previously I explained how the dots were generated

the squares that is

now I want you to explain that back to me, so that I know how much you gathered from my previous explanations

with the math looking like this, it goes without saying that you need to know the approach before you can read the schematic

[left gyro]

so tell me, how did I place these blocks?

what approach or general idea was I using?

you can be really broad here, all the specific details are dealt with later

[tulip plover]

continued fractions?

[left gyro]

thats not correct

try again

it involved something like a custom variant of the bresenham algorithm, didnt it
try mentioning something about that

you can also refresh your copy of the 2D link, there was a bug in the working of the math that I fixed

it should show properly now

lines 7 through 10 show part of the working

[left gyro]

parts of it are still convoluted as a result of getting it to work in desmos, so just general ideas are OK here

[tulip plover]

okay the y0 is trying to find the center of the block and as it has values of .5, 1.5, 2.5 I think and so the floor (y) means x is approaching the lower limit which then 1/2 gets added and the sqrt 3 is the slope of the equilateral so it is basically getting a rough estimate of the slope using that as the denominator? and then we are checking the x placement which if left 0 you would have something that determines the placement on the negative side since it is x - y0-(floor(y) + 1/2)/sqrt3 and then the other side is the same just it has a ceiling on it to it that gets added 1 {0<y<ceil(y0)} that basically determines the symmetrical side

[left gyro]

again its not recommended to attempt to read the purpose of the math from its schematic

youre taking the hard way around

you could also just look back on what we said earlier: #help-15 message

[tulip plover]

that channel was hidden?

is ovee

[left gyro]

bro

thats our previous help channel

[tulip plover]

yeah I know

but it was hidden yesterday

I couldn't read the messages

it came back

[left gyro]

have you tried just searching for your own messages

I searched up proportional to find my post

[tulip plover]

it was completely gone from the channel list I looked for it

[left gyro]

from: @matt07734 proportional to be specific

doing that skips whether a channel is hidden or not

[tulip plover]

hmm

[left gyro]

you might as well be saying you couldnt find a website because it was on the 2nd page of google

[tulip plover]

I was under the impression that help channels get purged

[left gyro]

oh I see

well if you look around you can see that that's not the case

but while I was looking, I noticed this must be your first time on this server

[tulip plover]

also isn't the .5 and the 1.5 the thing I was saying?

we were looking for the center of the squares

to fill in the blocks

[left gyro]

thats only part of it

Im not sure if I showed you what the floor(x) + 1/2 exactly did, I mightve done that

[tulip plover]

we try to find the center that is closest to the slope of the triangle

[left gyro]

thats part of the schematic that rounds to the nearest integer + 1/2

[left gyro]

closest to the line you mean

[tulip plover]

which is the slope of the triangle?

the line is the slope of the triangle?

[left gyro]

thats not how the word slope is used

the word slope means the rise/run of a given line, it is a number (√3), not a line

[tulip plover]

the triangle's side

yes

[left gyro]

yes there we go

thats the general purpose of the dots

[tulip plover]

I basically kind of said that from the schematics

[left gyro]

you tried to get into detail which has some issues in the way you attempted to explain it

[tulip plover]

hmm

[left gyro]

lets not do that until we piece this together

[tulip plover]

okay

[left gyro]

first, "approaching" is reserved for infinite processes like limits, try not to use it

[tulip plover]

the issue is that when I wanted a summary I wanted this talk of clear up to be out of it

[left gyro]

this what?

talk of clear up?

wdym

[tulip plover]

like we try to clear things up

like right now

[left gyro]

well we already had that once

[tulip plover]

it bloats it

[left gyro]

thats when I explained it to you

well lets do it again

[tulip plover]

I thought floor was limits?

[left gyro]

limit is a special math word that means something else

use a different word like bound

[tulip plover]

yes but I was thinking they were the same

[left gyro]

they are not the same, anyways as I was saying

what we did here was place a block at each 0.5, 1.5, 2.5

then round it to the nearest block, using either floor(x) + 1/2 or ceil(x) - 1/2, both do the same

now theres something here I dont think you noticed

[left gyro]

we wanted 2 things for our block placement

the first was that each block was intended to be closer to the line than other options

rounding can achieve that

but the second is that the line has no gaps

the line here has a slope > 1

that means we go up more than we go right

so we need a block placed at every height, at least 1

[left gyro]

heres a slightly stronger example

you can see here the middle block has some pretty high error from the line

but it must be there as not to create a gap in the line

this one idea is what greatly simplifies what we are doing

it now means we know exactly where to look when we need to place blocks

same thing for the 3D example:

this face if best facing towards the yz axis

its not the same example as I showed you earlier with the xz axis but I think you should see a diagonal face for a change

here, we know there should be a block at every (y, z) coordinate

so that lets us, similar to the 2D case, choose the nearest block of the particular "row"

unfortunately the exact schematic involved differs between 2D and 3D, because 2D is really fast at shading while 3D is not

the 3D is in fact missing the block shading, it just shows you the dots

getting those dots in the right place is a technical detail, we can ignore that for now because I want you to understand the 2D schematic first

[tulip plover]

is it done like where it's like if there is a y difference that is greater than one from the center of the previous block to the next block then a block should be placed on top of the previous block to close the gap

[left gyro]

no, as stated earlier it isnt the bresenham algorithm

lets get into the details

we will focus on three lines: 6, 7, 8

line 7 is the line we want to approximate with blocks
line 8 are the actual dots placed to do so

now the top of the outer triangle is at (X, y0)

where X is the x-coordinate of the centroid
and y0 is calculated to be the top of the outer triangle (as 3Y - 1 if I remember)

so we first have to generate a list of y-coordinates

first one is for the bottom most point

last one, ceil(y0) - 1/2, is for the top most point

as you can see here, the points generated start at 1/2 and end around at y0

from that list of y-coordinates,

we plug it into the equation to get the x-coordinate

then we ceil(...) - 1/2 it to round it to the nearest block

this is written as a point, with the x- and y-coordinates like that

that way desmos displays it as a list of points

for example the 3rd entry in Y0 is 2.5

heres an example of "line 7" and "line 8" for 2.5

matching up with:

so you get a y-coordinate, find the x-coordinate using the line equation, then ceil(...) - 1/2 to round to the nearest integer + 1/2

the result is a point at that y-coordinate closer than any other

do this for each to get the entire set of points

[left gyro]

distinctly notice here that, unlike the bresenham line, each coordinate is known in advance

bresenham's algorithm builds it up from the bottom or from the top, point by point

this just does each one with a formula

a variant of drawing the line that in some ways isnt very bresenham at all

bresenham is intended to be quick and cheap, this is quick but not cheap with that √3 floating-point calculation

besides, I mentioned earlier that desmos doesnt do well with for loops
this is essentially the easiest desmos way to get these points
regardless I wouldve done this even if it wasnt desmos due to how simple it can be done

do you understand so far?

[tulip plover]

hmm I see

[left gyro]

with this cleared up, this is when we can summarize

[tulip plover]

I am getting a little confused with the lower case y0

[left gyro]

capital letters for lists and constants

lowercase letters for everything else

[tulip plover]

it's just 3y-1 yes

[left gyro]

yes the letter conventions got shifted around when I polished it

y0 is calculated from Y to be the y-coordinate of the top of the triangle

the calculations shown are simplified, it is not immediately clear why it would just be 3Y - 1

you could view it as 3(Y - 1/2) + 1/2

and use that a centroid is at 1/3 height of a triangle

[tulip plover]

hmm alright

[left gyro]

so we have y0

then we get Y0, a list of points to draw the sides of the triangle

then we find where they are on the line

then we round them to the nearest block

exactly 1 point per y-coordinate, to leave no gaps

and we can use that because the line is more vertical than horizontal

[tulip plover]

per side yes

[left gyro]

whoops, yes thats true

from here are some other steps, but this is where the 2D and 3D versions diverge

lets go over to that since that was your question

the math here is a bit more involved, some dot and cross product was used to get this to work right

oh hm

youre familiar with dot and cross product?

[tulip plover]

last thing I did was calculus

I don't know

[left gyro]

what about determinants?

[tulip plover]

I have not done that

[left gyro]

thats alright

heres how you can see it:

[left gyro]

we need a version of line 7, which would be the face itself that we are trying to approximating

in 3D, this would be a plane

now have you seen any ways to write down a plane? for example x + y + z = 1 or z = x + y?

[tulip plover]

so the plane must have y, x, z equal a single value?

[left gyro]

its just the (x, y, z) for which that equation is true

it looks like a plane, beacuse there are two directions you can go in that still keep you on the plane

for example if you had x + y + z = 1,

you can head in the (1, -1, 0) direction and still be on the same plane

so for example if you started at (0, 0, 1)

(1, -1, 1) is still on the plane

(2, -2, 1) is still on the plane

(x, -x, 1) is still on the plane for any real x

so (1, -1, 0) is seen here as a direction that keeps you on the plane

as opposed to most directions which do not do this, they leave the plane

a similar idea is also true for lines:

if you add (1, √3) to a point, it will remain on the line

the difference between a line and a plane is that a line has 1 direction, a plane has 2

any other direction would be some addition of the 1 or 2

[tulip plover]

oh I see

[left gyro]

you can see here its incredibly general and yet simple which way it goes

[left gyro]

"1 direction" for a 2-directional line

because if you have the one direction (1, √3),

you can scale it by any amount and still be on the line

(2, 2√3)
(√3, 3)
(-1, -√3)
etc.

[tulip plover]

you are basically talking about that corner thing as the plane yes?

when you f3

or in modeling in blender

the red blue and green line

[left gyro]

the corner thing? as in the lower-left corner?

[tulip plover]

it has 3 different directions

for the x y and z axis

[left gyro]

the F3 crosshair?

[tulip plover]

yes

[left gyro]

I can confirm we are counting directions in that sense

3D for 3 directions

some amount of x + some amount of y + some amount of z

each direction an independent way to move a point somewhere

planes have 2 directions, lines have 1

for this reason, lines are 1 dimensional and planes are 2 dimensional

[left gyro]

two directions you can pick here are (1, -1, 0) and (1, 0, -1)

any other direction that keeps you on the plane is a sum of multiples of these directions

its not easy at first to prove that this is the case, so I wont do it for now

probably wont show it to you at all, its not too necessary

[left gyro]

but you do get the idea that these 2 directions would be all you need for this plane, right?

that and a starting point like (0, 0, 1) ig

[tulip plover]

yes

[left gyro]

very good

now the way I did this was suited for me, because I wanted a painless and quick way to get that plane

as a result, the math isnt readable, especially since youre not familiar with the dot or cross product

[tulip plover]

it's ovee

[left gyro]

however I can explain it roughly this way:

see these controls?

this was part of the process I was using to create the points

Im gonna have to fix this error then maybe itll be clearer for you

ok here we go

we know this plane faces most towards the yz axis

so we need to place exactly 1 block per y and z coordinate for this plane

the y3 and z3 sliders control this purple line which goes through the triangle

its coordinates allow for a variable called t

this t can vary between 1 and s
desmos graphs all of them
so you see a line that begins at (1, y3, z3) and ends at (s, y3, z3)

this line is parallel to the x-axis and pierces through the face

the question now is to determine this point, the point where the line intersects the plane

I find the coordinate of this point to be:

you can see the x-coordinate depends on the y- and z-

[left gyro]

the dot cross determinant stuff was just to find this point

once I have it, we are back in familiar territory

I dont know why I didnt do this earlier but Ive hidden the other faces

as before, we round the x-coordinate to the nearest block

floor(...) + 1/2

look closely and see that it is similar to the 2D version's line 8

final thing Id like to point out: desmos does not have such a thing as lists in lists

so you have to use a special command called for to plot a grid of points

[tulip plover]

hmm

[left gyro]

or at least to use a 2D grid of values

y3 being [.5, 1.5, ..., s]

and z3 also being [.5, 1.5, ..., s]

for will use this to give you:
(.5, .5), (1.5, .5), ..., (s, .5),
(.5, 1.5), ...

goes through the first coordinate, then goes through the second one

in the schematic this list is stored as F3
for instance, F3[16] would be the 16th point

you can see here the y and z coordinates are (15.5, .5)

[left gyro]

lines up with the list here

(.5, .5), (1.5, .5) ... (15.5, .5) ...
1st 2nd 16th

does this make sense?

[tulip plover]

the y part makes sense but I'm not seeing the z part

is the z working like this

where it turns to this side to make a another 2d like triangle going up?

[left gyro]

thats rather vague, I cant say if you got it or not

for this points for example,

[tulip plover]

I am saying is it doing the same thing we did to find the 2d triangle

just with y and z

like how in the video the guy used hull to make the 3d shapes in the mod

[left gyro]

this list of points acts the same as Y0 in being a set of coordinates to each place a block

[tulip plover]

he pulled the z axis up

[left gyro]

ok for these points for example,

the first four points are these at the bottom

the bottom-right corner of this image is at (.5, .5, .5), that is the 1st point, F3[1]

so the bottom row is F3[4], F3[3], F3[2], F3[1], left-to-right, from this angle

[tulip plover]

visually confusing without a grid

[left gyro]

this should be clearer

[tulip plover]

I can see it but where is the middle of the block supposed to be the balls are at the center?

[left gyro]

theres a bit of a necker cube happening here

ok this angle should be good

we are currently looking at the inside of the face

you can see the y- and z- directions

at the bottom row, from left-to-right, are the points F3[1], F3[2], F3[3], and F3[4]

the y- and z- coordinates are (.5, .5), (1.5, .5), (2.5, .5), (3.5, .5)

[tulip plover]

so the preplotted dots like in the 2d one I just realized it's very small which is why this happens

[left gyro]

?

[tulip plover]

aren't these the dots that have to be connected like how you did the 2d one?

[left gyro]

oh I see

yes Im showing you a very small example to better understand F3's behavior

there are very few dots here

each dot still represents a block

[tulip plover]

yes

[left gyro]

also, theres a slight bug now that Im looking at this

[.5, 1.5, ... s] ends up being 5 points here due to desmos rounding s I think

or something like that not too sure

Ive fixed it as this

when s is 4, there are 4 points per row

ok as I was saying

F3 then has F3[5], F3[6], F3[7], F3[8] go like this

(.5, 1.5), (1.5, 1.5), (2.5, 1.5), (3.5, 1.5)

you can see here the for command is going through the grid of points one coordinate at a time

first it goes through every first coordinate

then it advances to the next second coordinate and goes through the first coordinates again

does this make sense?

[tulip plover]

this right

[left gyro]

you dont actually need the outer []s if youre viewing that all by itself

but thats the correct syntax

try doing (x, y, 1) for x = [1, 2, 3], y = [1, 2, 3]

put in L = before it to define L as this list

then do L[1], L[2], etc. to see how this list goes through every point

does the order make sense?

[left gyro]

the line automatically displays the stuff on the right side of the screen as you type stuff

you can just quickly go through the numbers 1 through 9 inside the []s on line 2

makes it easier to see the behavior

[tulip plover]

I see

[left gyro]

Im going to pretend that was a video of you going through all 9 points

does this order make sense

[tulip plover]

I don't get it

after 3 it starts going into the green triangle

[left gyro]

I think you should open a new desmos.com/3d tab

instead of trying to look at it from afar and upside down

https://desmos.com/3d

and type in L = (x, y, 1) for x = [1, 2, 3], y = [1, 2, 3]

[tulip plover]

oh I think I see it now

these are the preplotted points

like a grid

[left gyro]

ok that doesnt instill trust in me but sure

[tulip plover]

is ovee

[left gyro]

you can at least see the order that the for command goes through them

through this, yes I have the grid of points

so for here,

y3 and z3 are going through 0.5, 1.5, etc.

and the x-coordinate is calculated based on those

as a result, I get a grid of points, with a point for every (y3, z3)

[tulip plover]

I see

[left gyro]

this is a technical desmos detail to see the for and its order

but I think its important you can at least read off the coordinates of the points

[left gyro]

if you wanted to do this manually to find a block for the face, you can for example put in y3 = 4.5, z3 = 5.5

into this

[tulip plover]

is still no working is ovee

[left gyro]

Im not even done explaining

I dont even know what youre referring to

whats not working

[tulip plover]

you said you fixed it

earlier at the start

[left gyro]

unfortunately I didnt get around to doing that yet

[tulip plover]

this

[left gyro]

because we're going to do it together

[tulip plover]

oh okay

I thought was finished that is why you were explaining

[left gyro]

nah I was gonna try and get you to be able to understand this

[tulip plover]

I see

[left gyro]

unfortunately since you dont know dot or cross product,

[tulip plover]

it's so ovee

[left gyro]

the actual fix looks to be beyond your skill, so the best I can do here is walk you through how it works

then Ill just go fix the problem

[tulip plover]

too much skill issue I'm crying

[left gyro]

youll sort of see what the problem is actually

[tulip plover]

hmm

[left gyro]

here's face 4

[tulip plover]

it also needs to account for the double tip

[left gyro]

less of that

[tulip plover]

hmm

[left gyro]

you notice here how the blue points are uh

merging into the other ones?

[tulip plover]

yes

[left gyro]

thats because I dont actually do a new calculation to get the face 4 points

[tulip plover]

I was curious about that

[left gyro]

I just uh

reflect the face 3 points

[tulip plover]

it's so ovee

[left gyro]

this only works if s is an integer

now I actually gotta do the face 4 points for real

[left gyro]

the .x, .y, .z get the respective coordinates out of F3

you can see all I do is do s - the x-coordinates then thats it

literal reflection

ok now that you know that, its time for the fix

[tulip plover]

so much win

[left gyro]

this might take a bit but maybe if I say enough words, you can understand it

I dont expect you to understand making face 4 though, sort of just watch the fireworks

[tulip plover]

I will never be a voxel tetrahedron designer

[left gyro]

you just havent learned vectors yet

its alr

go learn vectors later on then you can maybe return to this

theyre relatively simple, but its a bit much to just know for one singular use

[tulip plover]

what level of math is that?

[left gyro]

high school or college level

[tulip plover]

calculus bc?

[left gyro]

no

vectors arent calculus

theyre like 2D point math

[tulip plover]

physics

[left gyro]

but in 3D

physics will go over this, yes

[tulip plover]

I did physics c

[left gyro]

oh I see

those dont go over this

well thats a shame

[tulip plover]

is ovee

[left gyro]

ok as before, we already know we can do floor(...) + 1/2 or ceil(...) - 1/2

and that we need a block per y- and z- coordinate

the only missing step is getting that face

face 4 is currently being displayed by desmos as between these three points

V is storing where the four vertices are

V[2] is the lower-left one near the x-axis
V[3] is the lower-right one with a high y-coordinate
V[4] is the top one

same as before:

y4 goes from 0.5 to ceil(s) - 0.5

oh thats strange

[tulip plover]

hmm

[left gyro]

ok for now, Im changing the y4 and z4 upper bounds to be these

instead of ceil(s) - .5

Ill need to think more on whether I needed to do this or not, but for now thats what it ends on

ok anyway for a (y4, z4) I have a line going through the triangle

and i need to find the coordinates of the one point that goes through the triangle

ok heres something I think you might understand

we know the coordinates of V2, V3, and V4

so lets consider two directions,

one from V[2] to V[3]

another from V[2] to V[4]

each direction is by definition the edge of this triangle

so theyre along the plane

now the dot cross whatever allows us to tell if a particular point is on a plane or not

we simply find the one point which would be on the plane

[tulip plover]

hmm so you do that all over the plane to make the blocks?

[left gyro]

yes

[tulip plover]

so truee

[left gyro]

after I find the point,

[tulip plover]

is ovee

[left gyro]

lol

hold on I need to actually find the point first before I say what happens after

we can first use something called the cross product

this takes in 2 vectors, and outputs a vector that is perpendicular to the both of them

for example, (1, 0, 0) cross (0, 1, 0) is (0, 0, 1)

unfortunately its a bit inconvenient that the vector goes out of frame

but this would be how itd look

what do I use this vector for? I can use a dot product

the dot product takes in two vectors and can tell you whether those vectors are perpendicular or not

perpendicular vectors always have a dot product of 0

blue dot green here is always 0 for either of the blue vectors

[tulip plover]

hmm

[left gyro]

now think about if we had a point on the purple line that was also on the triangle