But this is like saying that it is interesting that two of the three number are prime because the density of prime numbers goes down as you go to larger numbers.

Or that it is interesting the 4 is a perfect square because it is so rare among integers for one to be a perfect square. After all, only 9% of the positive integers less than 100 is a perfect square while only but about 1% of the integers less than 10,000 is and only 0.1% of the integers less than one million is.

There are infinitely many Pythagorean triples (i.e., integer solutions to the Pythagorean equation). Again, plug ANY two different positive integers m and n into the formulas I gave and you have a Pythagorean triple. If m = 2 and n = 1, you have 3,4,5.

So what you are really saying is that it is somehow interesting that if you take the difference of the squares of two integers you get an integer, or that twice the product of two integers is an integer, or that the sum of the squares of two integers is an integer. But, to me, that seems pretty much like the only thing that can happen, so not very interesting.

If anything, what I would say would be somehow interesting is that there is no Pythagorean triple that consists of three prime numbers -- but it's only interesting until you look at the equations I gave you and you see that this has to be true (provided that all primitive Pythagorean triples can be generated by those equations using a suitable choice of m and n, which they can).

 

Hi,

Yeah but i am not sure that you realize that you are arguing a point that is highly subjective subject to individual appreciation of beauty or aesthetics. You may not find it interesting but i think that out of 100 possibilities only 2 (or 4) work so that is surely more interesting than if all 100 worked.
One possible implementation might be to plot all solutions as points on the x,y plane and connect the dots. However even that may be limiting the possibilities of how it might be interpreted as beautiful.
As another example if this was a physical construction we could build a triangle without cutting any pieces if we had only unit pieces to work with. Maybe 3 units high, 4 units long, and 5 units connecting the two free ends is one possible construction from the set of solutions.

I am guessing that maybe you dont think that the solution to:
x^2-x-1=0

is also not interesting just because it's just an algebraic equation of which there are many.

 

Hi,

Yeah but i am not sure that you realize that you are arguing a point that is highly subjective subject to individual appreciation of beauty or aesthetics. You may not find it interesting but i think that out of 100 possibilities only 2 (or 4) work so that is surely more interesting than if all 100 worked.

If all 100 worked, then why wouldn't that be interesting in it's own right?

One possible implementation might be to plot all solutions as points on the x,y plane and connect the dots. However even that may be limiting the possibilities of how it might be interpreted as beautiful.

We don't have to guess too much about what that would look like. For each primitive solution all solutions that are integer multiples of those values are also Pythagorean triples, so you have an infinite number of points that lie on the radial line extending from the origin through the primitive solution. Now draw an arbitrary radial line out from the origin. You can find a Pythagorean triple that lies arbitrarily close to that line.

I am guessing that maybe you dont think that the solution to:
x^2-x-1=0

is also not interesting just because it's just an algebraic equation of which there are many.

Why should it be "interesting" just because the golden ratio is one of the solutions?

The golden ratio is a number that comes about from looking at certain problems, mostly recursive problems. It's not a stretch to consider it an interesting number, or at least a number that, like pi and e and many other numbers, comes about from looking at the behavior of various things.

It happens to have a form that makes it trivial to construct an algebraic equation for which it is one of the solutions.

Why does that make the constructed algebraic solution interesting?

This is like saying that it is "interesting" that if you take half of the square of the diagonal of a square that you get the area.

I'm much more likely to consider an equation like

1 + e^(jπ) = 0

as "interesting" because it involves five of the most fundamental constants together so tightly. But even then I would be more likely to call it "elegant" as opposed to interesting, because as soon as you discover that

e^(jθ) = cos(θ) + j sin(θ)

equations like that become trivially easy to construct.

 

insane stuff

{┌──────────────────────────┐}
{│  256 Color Bitmap Maker  │}
{│ ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀ │}
{└──────────────────────────┘}
{════════════════════════════════════════════════════════════════════════════}

program _256maker_liz;

{════════════════════════════════════════════════════════════════════════════}

uses crt;

{════════════════════════════════════════════════════════════════════════════}

const

  HorizPicSz : word =  688;
  VertiPicSz : word = 1024;
  ImgFlNme : string = 'E:\liz_FGh.bmp';

{════════════════════════════════════════════════════════════════════════════}

type

  art = array[0..1023]of byte;

  fht = array[0..$35]of byte;

{════════════════════════════════════════════════════════════════════════════}

const

  Fsz : byte = $02;
  Cls : byte = $12;
  Rws : byte = $16;

     Fl256Hdr : fht =
{00}($42,$4D,$00,$00, $00,$00,$00,$00, $00,$00,$36,$04, $00,$00,$28,$00,
            { Total FileSize }
{10} $00,$00,$00,$00, $00,$00,$00,$00, $00,$00,$01,$00, $08,$00,$00,$00,
            {Columns}        {  Rows }
{20} $00,$00,$A0,$0F, $00,$00,$00,$00, $00,$00,$00,$00, $00,$00,$00,$01,

{30} $00,$00,$00,$01, $00,$00);


{════════════════════════════════════════════════════════════════════════════}

procedure click;
begin
sound(880);
delay(18);
nosound
end;

{════════════════════════════════════════════════════════════════════════════}

function pict( x, y : integer ) : byte;
var
  aa, bb,
  xx, yy,
      cc : double;
    c, i : integer;
function power( arg : double;
                 pow : integer ) : double;
begin
if arg < 0
then power := ( 1 - ( pow mod 2 ) shl 1 ) *
                    exp(ln(-arg)*pow)
else if arg = 0
      then power := 0
      else power := exp(ln( arg)*pow)
end;
begin
  xx := (x - HorizPicSz/2-0.51)/13*512/5/HorizPicSz;
  yy := (y - VertiPicSz/2-0.53)/13*512/5/HorizPicSz;

  cc := 0;

  for i := 1 to 13
  do cc := cc + (power(cos(xx*Sin(yy)*i),i)
                *power(sin(yy*Cos(xx)*i),i));


  c := round( 16*(2*(cc + 16) - 24) );

  if c > 0
  then begin
            c := -ord(c>255) and $FF or c;
         pict := Lo(c)
       end
  else pict := 0
end;

{════════════════════════════════════════════════════════════════════════════}

function pal256( rg : word ) : byte;
var
  ck : word;
begin
  ck := rg mod 4;
  rg := rg div 4;
  {$R-}
  if rg > 0
  then case ck
       of 1: if rg < 85
             then pal256 := 85+2*(85-rg)
             else if rg > 170
                  then pal256 := 85+2*(rg-170)
                  else pal256 := 0;

          2: if( rg > 85 )and( rg < 171)
             then pal256 := 85+2*(rg-85)
             else if( rg > 170 )and( rg < 255 )
                  then pal256 := 85+2*(255-rg)
                  else pal256 := 0;
          3: if rg < 86
             then pal256 := 85+2*rg
             else if rg < 170
                  then pal256 := 85+2*(170-rg)
                  else pal256 := 0
        else pal256 := 0
       end
  else pal256 := 0
  {$R+}
end;

{════════════════════════════════════════════════════════════════════════════}

var

    bmp_fl : file of byte;

    btt : byte;

  scln : ^art;

  flsz_X, flsz_Y,

  flsz_Total : longint;

       dwp : array[0..1]of word absolute flsz_Total;

  ctx, cty,
  x_size, y_size,
       bxs : word;

{════════════════════════════════════════════════════════════════════════════}

begin

  clrscr;

  flsz_X := 0+HorizPicSz;
  flsz_Y := 0+VertiPicSz;
  flsz_Total :=  $36 + $40 + flsz_X*flsz_Y;

  bxs := HorizPicSz;
  bxs := bxs - 1;

  Fl256Hdr[Cls  ] := Lo(HorizPicSz);
  Fl256Hdr[Cls+1] := Hi(HorizPicSz);

  Fl256Hdr[Rws  ] := Lo(VertiPicSz);
  Fl256Hdr[Rws+1] := Hi(VertiPicSz);

  Fl256Hdr[Fsz  ] := Lo(dwp[0]);
  Fl256Hdr[Fsz+1] := Hi(dwp[0]);
  Fl256Hdr[Fsz+2] := Lo(dwp[1]);
  Fl256Hdr[Fsz+3] := Hi(dwp[1]);

  new(scln);

  Assign(bmp_fl,ImgFlNme);
  Rewrite(bmp_fl);

  {$R-}

  for cty := 0 to $35
  do write( bmp_fl, Fl256Hdr[cty] );{ write header }

  for cty := 1 to $400
  do begin
       btt := pal256(cty);
       write( bmp_fl, btt )
     end;                   { write palette }

  {$R+}

  for cty := VertiPicSz-1 downto 0
  do begin
       {$R-}
       for ctx := 0 to bxs
       do scln^[ctx] := pict(ctx,cty);

       for ctx := 0 to bxs
       do write(bmp_fl,scln^[ctx]);
       {$R+}

       GotoXY(1,1);
       Write((100*(1-(cty/VertiPicSz))):3:2,'% done')

     end;

  Close(bmp_fl);
  dispose(scln);

  Writeln;
  Writeln('... Picture Done.')

end.

{════════════════════════════════════════════════════════════════════════════}

!basically! the color - the variable cc - is got by (it is then tweaked a bit from to fit 16 or 256 color palette)

cc := 0; for i := 1 to 13
do cc := cc + (power(cos(xx*Sin(yy)*i),i)*power(sin(yy*Cos(xx)*i),i));

 

If all 100 worked, then why wouldn't that be interesting in it's own right?

Well now you are arguing that nothing is interesting unless you believe it is.

Usually what makes something interesting is it's rarity, but yes all inclusive could become interesting if there was some unusual fact that would make one believe the opposite without looking more closely.

If there was a ship with 100 18 year old males on it and one female came on board that would be interesting to the males, but it could also be interesting to the female that there were 100 males unless she was nervous about it <smile>.


Smilies stopped working for some reason.

Why should it be "interesting" just because the golden ratio is one of the solutions?


The golden ratio is a number that comes about from looking at certain problems, mostly recursive problems. It's not a stretch to consider it an interesting number, or at least a number that, like pi and e and many other numbers, comes about from looking at the behavior of various things.


It happens to have a form that makes it trivial to construct an algebraic equation for which it is one of the solutions.


Why does that make the constructed algebraic solution interesting?


This is like saying that it is "interesting" that if you take half of the square of the diagonal of a square that you get the area.


I'm much more likely to consider an equation like


1 + e^(jp) = 0


as "interesting" because it involves five of the most fundamental constants together so tightly. But even then I would be more likely to call it "elegant" as opposed to interesting, because as soon as you discover that


e^(j?) = cos(?) + j sin(?)


equations like that become trivially easy to construct.

Again it sounds like you are arguing that nothing is interesting unless it is interesting to you.
I think something is interesting if it is rare or a rare occurrence.
Also if something is new to us it may be interesting which means it is rare even if just for us at the time.
The number 'pi' is more interesting than many other numbers so is the number 'e'.
To find out more about how integers can be more interesting than other numbers read about the mathematician who was a master of integers: Srinivasa Ramanujan

 

Well now you are arguing that nothing is interesting unless you believe it is.

No more so than you are -- and that's largely the point. As you say, what is interesting is so subjective that it becomes quite personal. So if one person is free to say that something is "interesting" isn't another person free to say that it isn't? Or does everyone have to just accept that something is "interesting" just because there exists someone that thinks it is?

That's why I originally asked why you found the fact that 3,4,5 is a Pythagorean triple to be "interesting" instead of just accepting that it was because you stated that it was. Was it because Pythagorean triples exist at all? Was it because it's a Pythagorean triple consisting of three consecutive integers? Was it some other reason? I was hoping that you had a reason that I might find interesting, too. I could probably get behind the latter because three consecutive integers that are a Pythagorean triple is not only rare, but unique -- and as far as I know not guaranteed to exist at all (at least until you specifically ask the question and work the math to show that there is a single, unique such triple). But it turns out that you find it interesting just that Pythagorean triples exist at all because they are "rare". But most things in math are "rare". Should people find the number 17 interesting just because it is a prime number and prime numbers are "rare"? Should people find that fact that (1/2) + (1/3) + (1/6) = 1 because having three non-integer rational numbers sum to an integer is "rare"?

If "interesting" is subjective, then don't be surprised when you get pushback about whether something that you find interesting really is. Take it as an opportunity to explore and share and discuss what you find to be interesting about it, but be prepared that you might not succeed. But whether you succeed or not, you are still perfectly free to continue to think of it as interesting -- in fact you may well find it even more interesting than you did before.

 

No more so than you are -- and that's largely the point. As you say, what is interesting is so subjective that it becomes quite personal. So if one person is free to say that something is "interesting" isn't another person free to say that it isn't? Or does everyone have to just accept that something is "interesting" just because there exists someone that thinks it is?

That's why I originally asked why you found the fact that 3,4,5 is a Pythagorean triple to be "interesting" instead of just accepting that it was because you stated that it was. Was it because Pythagorean triples exist at all? Was it because it's a Pythagorean triple consisting of three consecutive integers? Was it some other reason? I was hoping that you had a reason that I might find interesting, too. I could probably get behind the latter because three consecutive integers that are a Pythagorean triple is not only rare, but unique -- and as far as I know not guaranteed to exist at all (at least until you specifically ask the question and work the math to show that there is a single, unique such triple). But it turns out that you find it interesting just that Pythagorean triples exist at all because they are "rare". But most things in math are "rare". Should people find the number 17 interesting just because it is a prime number and prime numbers are "rare"? Should people find that fact that (1/2) + (1/3) + (1/6) = 1 because having three non-integer rational numbers sum to an integer is "rare"?

If "interesting" is subjective, then don't be surprised when you get pushback about whether something that you find interesting really is. Take it as an opportunity to explore and share and discuss what you find to be interesting about it, but be prepared that you might not succeed. But whether you succeed or not, you are still perfectly free to continue to think of it as interesting -- in fact you may well find it even more interesting than you did before.

That's not true. You are saying mine is not interesting i am NOT saying that yours is not interesting but i am saying that there is a difference sometimes. But dont take my word for it, look it up.
What i find amazing is that you are still arguing this point. It does not really matter to me what someone else thinks about what i think. If i think something is interesting, that's just too bad if someone else doesnt when it comes down to the brass tacks. Beauty is in the eye of the beholder.
I might think that the number 2 is more interesting than any other number from 2 through 12. Sometimes called "Cat's Eyes" but if someone else thinks 7 is more interesting because it wins something on the first try that's their opinion. I might agree or disagree, too bad if a 3rd person doesnt like either.

 

I have learned this since from my schooling to engineering in physics and maths.
We used to remember them by some tricks like All Silver Tea Cups.

 

I have learned this since from my schooling to engineering in physics and maths.
We used to remember them by some tricks like All Silver Tea Cups.

Oh i presume that is the mnemonic to remember the quadrants but you might take a minute to explain that to the readers so they can use that too it's not bad at all.

 

hi,
One rhyme I have seen used for Sin Cos Tan is:
Peter Has Been Here Playing Billiards

E

 

We learned sin, cos and tan a little differently in England, back in 1950 but it was very effective because I always remember:
Sign (sin) of a public (perpendicular) house (hypotenuse). Sine = perpendicular / base
Costly (cos) black (base) hat (hypotenuse). Cosine = base / hypotenuse
Tan (tan) a poor (perpendicular) boy (base). Tangent = perpendicular / base

 

This is one of my easy to remember long time favorites:

Antidisestablishmentarianism
Spectrophotofluorometrically
Thyroparathyroidectomized
Compartmentalization

although i like to make it even easier to remember:
Antidisestablishmentarianism
Spectrophotofluorometrically
Thyroparathyroidectomized
Cat

<chuckle>

 

The best way to remember anything is through mnemonics. Make the word into a story

Mnemonics Dictionary would have something I bet.