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Pythagorean triples

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  • Neil Fernandez
    Hi Mike and everyone. I have been playing/working with Pythagorean triples. Here is the list of all 14 triples where each number is
    Message 1 of 6 , Jul 26 12:29 PM
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      Hi Mike and everyone. I have been playing/working with Pythagorean
      triples. Here is the list of all 14 triples where each number is <= 100,
      *and* the triple is not a 'multiple' of another triple. For example, 6 8
      10 is not included because it is a multiple of 3 4 5, and gives the same
      triangle as 3 4 5. (I say 'same' rather than 'similar', because we
      haven't brought units of length into consideration, so it is not just
      similarity).

      These triples could be considered as 'prime', in the sense that they
      aren't 'multiples' of any smaller triples. How close a connection there
      might be with Gaussian primes, I don't know. In what follows I refer to
      ones that are not prime as 'composite'.

      3 4 5
      5 12 13
      7 24 25
      8 15 17
      9 40 41
      11 60 61
      12 35 37
      13 84 85
      16 63 65
      20 21 29
      28 45 53
      33 56 65
      36 77 85
      39 80 89

      Here is the longer list of all triples with numbers <= 100, including
      both prime and composite triples:

      3 4 5
      5 12 13
      6 8 10
      7 24 25
      8 15 17
      9 12 15
      9 40 41
      10 24 26
      11 60 61
      12 16 20
      12 35 37
      13 84 85
      14 48 50
      15 20 25
      15 36 39
      16 30 34
      16 63 65
      18 24 30
      18 80 82
      20 21 29
      20 48 52
      21 28 35
      21 72 75
      24 32 40
      24 45 51
      24 70 74
      25 60 65
      27 36 45
      28 45 53
      28 96 100
      30 40 50
      30 72 78
      32 60 68
      33 44 55
      33 56 65
      35 84 91
      36 48 60
      36 77 85
      39 52 65
      39 80 89
      40 42 58
      40 75 85
      42 56 70
      45 60 75
      48 55 73
      48 64 80
      51 68 85
      54 72 90
      57 76 95
      60 63 87
      60 80 100
      65 72 97


      You can get a nice pattern if you do a spreadsheet with 1-100 across the
      columns and 1-100 down the rows, and a '*' where sqrt(L^2 + M^2) is a
      perfect square.

      Give me a few days and I will send some nice spreadsheets on this to the
      list, including one based on Mike's with angle sizes added as asked for.

      Here is Mike's two-parameter formula again.
      L = 2PQ
      M = P^2 - Q^2
      N = P^2+ Q^2

      Here are the one-parameter ones I mentioned. The first is for L odd, the
      second is for L even.

      L=P
      M=((P^2)-1)/2
      N=((P^2)+1)/2
      for odd P (>1)

      and
      L=2P
      M=(P^2)-1
      N=(P^2)+1
      for even numbers 2P (again, P>1)

      Note that for odd L, M and N add up to L^2, and for even L, M and N add
      up to L^2/2. For odd L, M and N differ by 1, or even L, M and N differ
      by 2. You get the following triples. I have put 'composite' triples in
      parentheses, and again only include ones where numbers <= 100.

      L M N

      3 4 5 4+5 = 9
      (4 3 5) 3+5 = 16/2
      5 12 13 12+13 = 25
      (6 8 10) 8+10 = 36/2
      7 24 25 24+25 = 49
      8 15 17 15+17 = 64/2
      9 40 41 40+41 = 81
      (10 24 26) 24+26 = 100/2
      11 60 61 etc.
      12 35 37
      13 84 85
      (14 48 50)
      16 63 65
      (18 80 82)

      The ones in the prime list not included here are:

      20 21 29
      28 45 53
      33 56 65
      36 77 85
      39 80 89


      Of these, the following are generated by Mike's formulae (with inputs as
      shown):

      P Q
      20 21 29 5 2
      28 45 53 7 2
      33 56 65
      36 77 85 9 2
      39 80 89

      That's enough for now.

      More later.

      Best regards,

      Neil
      --
      Neil Fernandez
    • Terrie Halprin
      Hi Neil! Oh, man! I love this math! It s incredibly BEAUTIFUL -in, like, every direction. I must ve spent half the day yesterday just staring at some of
      Message 2 of 6 , Jul 27 9:35 AM
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        Hi Neil!

        Oh, man! I love this math! It's incredibly BEAUTIFUL -in, like, every
        direction. I must've spent half the day yesterday just staring at some of
        those numbers/symetries, on the whole and by the strand. I especially love
        how you've fashioned this concept into two solid pieces AND created two good
        mirrors/grids for us all to be seeing: it's like looking at a microcosmic
        prime and a macrocosmic whole -oh, yeah, it's beautiful!

        Today, I was thinking about that micro prime -that it'd be interesting to
        take all of those numbers and add/multiply(?) them up 1by1, 2by2 and/or 6by6
        then divide/substract(?) them all out by that 14 -and/or, maybe the 3.?
        It's so exciting! No doubt some very interesting infos gets stashed amongst
        a golden order like this. I also like the idea of working (playing
        between) these two mirrors with a few different modes/operators. So many
        options, so much food for thought!

        Gosh, yeah, am I looking forward to those spreadsheets (when they come up,
        of course,) -also- I wonder, will you illustrate (geomathematically) that
        primary grid? That'd probably be so beautiful to look at!

        Have a BEAUTIFUL day,

        Terrie

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      • Neil Fernandez
        In message , Terrie Halprin writes ... I think so too! On the 100x100 spreadsheet I mentioned,
        Message 3 of 6 , Jul 27 10:02 AM
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          In message <LAW2-F122bAUfKRfcjc00000240@...>, Terrie Halprin
          <thalprin@...> writes
          >Hi Neil!
          >
          >Oh, man! I love this math! It's incredibly BEAUTIFUL

          I think so too! On the 100x100 spreadsheet I mentioned, you get lines of
          asterisks going off, each one corresponding to the set of multiples of a
          prime triple, e.g. for (3,4,5), (6,8,10), (9,12,15) etc. As with many
          things in number theory, you get a combination of patterns and
          'randomness' in an uncountable number of 'directions', but of course all
          based on an underlying simple truth.

          It would be nice to bring colour variation into an expression of this.

          >it's like looking at a microcosmic
          >prime and a macrocosmic whole -oh, yeah, it's beautiful!

          It would even be possible to get a single constant out of it, for
          example by making a three-dimensional array, consisting of points
          (x,y,z) where x,y,z are all non-negative integers. Where they form a
          Pythagorean triple, put a '1', where they don't, put a 0. Then study the
          'spread' of 1s as you get further away from the origin, finding a single
          number to express how the spread varies.

          >Today, I was thinking about that micro prime -that it'd be interesting to
          >take all of those numbers and add/multiply(?) them up 1by1, 2by2 and/or 6by6
          >then divide/substract(?) them all out by that 14 -and/or, maybe the 3.?

          To try to whittle them down/seek an underlying order? I think as one got
          deeper with that one would have to go outside the set of integers.

          Another thing that would be interesting would be to study the set of
          triples which are not generable by the two formulae, which starts:

          33 56 65
          39 80 89

          >It's so exciting! No doubt some very interesting infos gets stashed amongst
          >a golden order like this.

          Best regards,

          Neil
          --
          Neil Fernandez
        • Terrie Halprin
          Hi Neil, Thinking about that micro prime -that it d be interesting to take all of those numbers and add/multiply(?) them up 1by1, 2by2 and/or 6by6 then
          Message 4 of 6 , Jul 27 11:29 AM
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            Hi Neil,

            Thinking about that micro prime -that it'd be interesting to
            take all of those numbers and add/multiply(?) them up 1by1, 2by2 and/or 6by6
            then divide/substract(?) them all out by that 14 -and/or, maybe the 3.?

            <To try to whittle them down/seek an underlying order? I think as one <got
            deeper with that one would have to go outside the set of integers.

            Yeah! Check it out: for me, when i see a grid like this maybe one of the
            first things that I always wonder is what it all adds up to -I want to know
            its quanity value -and, yes, that's mathematical space, organized chaos,
            surely, but, for all we know it's well worth exporing, maybe, it yeilds the
            revolution of Saturn, who knows, maybe something, maybe nothing, though I
            doubt it cuz it's a beautiful grid -SO- I like to take a look to see what
            the entire numerical expression values at (like one giant breath) AND then I
            like to think about what that means, and, maybe, it's an adventure -THEN-
            like in this instance I like to take this kind of quanity value and put it
            back into it's grid by way of accounting for all of its actual positions
            (exhaling it back out) -here being that there are 14 3s and/or 14 6s
            (strands) I'd probably divide the sum out by 42 and/or 84 -I'd make pizza
            dough out of the grid throw it up in the air with a great big swirl then
            plop it right down into its pie pan -just to see, to make another mirror out
            of base/averaged porportions and compare the two.? I'm an incredibly
            curious young lady! What fun!

            <Another thing that would be interesting would be to study the set of
            <triples which are not generable by the two formulae, which starts:

            33 56 65
            39 80 89

            Yeah! This is interesting. It's like the Super Mario Brothers movie -like
            trying to put a chard back into a crystal Oooh, I like that! What fun!
            This math, it just gets more and more interesting.

            Have a BEAUTIFUL day,

            Terrie

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          • Terrie Halprin
            Hi Y all, Oh, man! Neil, I am having such a blast with this grid that you ve fashioned! Playing around mostly (as yet) with the micro prime grid AND -oh,
            Message 5 of 6 , Jul 29 2:18 AM
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              Hi Y'all,

              Oh, man! Neil, I am having such a blast with this grid that you've
              fashioned! Playing around mostly (as yet) with the micro prime grid AND
              -oh, yeah, the night fishing is good!

              (quanity/perimeter order)
              1. 3 4 5 =12 (1) 1. 3 4 5
              2. 5 12 13 =30 (2) 2. 5 12 13
              3. 7 24 25 =56 (4) 3. 8 15 17
              4. 8 15 17 =40 (3) 4. 7 24 25
              5. 9 40 41 =90 (7) 5. 20 21 29
              6. 11 60 61 =132 (9) 6. 12 35 37
              7. 12 35 37 =84 (6) 7. 9 40 41
              8. 13 84 85 =182 (12) 8. 28 45 53
              9. 16 63 65 =144 (10) 9. 11 60 61
              19. 20 21 29 =70 (5) 10. 16 63 65
              11. 28 45 53 =126 (8) 11. 33 56 65
              12. 33 56 65 =154 (11) 12 13 84 85
              13. 36 77 85 =198 (13) 13. 36 77 85
              14. 39 80 89 =208 (14) 14. 39 80 89
              _______
              =1526 divided by 14=109

              I took the triples grid and summed up each of the individual triples to take
              a look at their quanity/perimeter values -then, I organized their
              arrangement < to > according to these sums -then, I listed the grid of
              triples (1-14) in their perimeter values order -I like how this organizes
              the hypotenuse. I considered looking into the 7triples with a
              quanity/perimeter sum under 100 (12,30,40,56,90,84,70=382) as kind of a
              sub-set AND 1526-382=1144. So, the sum splits up nicely into two halfs (2
              sevens, 382 and 1144). I like the 382 cuz 3+8+2=13 -so, it relates nicely
              to the 5,12,13 triple and the 1144 11+44=55 relates well with the 5 in the
              same (5,12,13) triple. This triple just gets more and more interesting!

              The 1526 divided by 14 =109, that's an interesting averaged out because
              mathematically it's right there, right next to the 108 degree trisection
              (here, there'd be 14 109s) -Also, because it's 109 it is itself probably
              naturally shadowed by a real interesting 110 -which would be interesting to
              align! You know, there's just one more triple I'd like to mention, if i
              may, in context to this grid -20,99,101 (=220) cuz 1526+220 =1746 -WOW(!)
              it's 1746 -the number of the celestrial marriage, a number which consists of
              both the feminine 1080 and the masculine 666 -that's just sooo beautiful!
              Imagine, this little triple rising up over these horizons/this lovely grid
              -here, especially so because of the 20,99,101, the 101 is the only number
              over 100 (over by one) -AND- this, it's just magic -It's the number (101) of
              the Arch Angel Michael. Man! Do I like that imagery!

              Next, I looked into area:

              3, 4, 5 =6sq
              5, 12, 13 =30sq
              8, 15, 17 =60sq
              7, 24, 25 =84sq
              20, 21, 29 =210sq
              12, 35, 37 =210sq
              9, 40, 41 =41sq
              28, 45, 53 =630sq
              11, 60, 60 =330sq
              16, 63, 65 =504sq
              33, 56, 65 =924sq
              13, 84, 85 =546sq
              36, 77, 85 =1386sq
              39, 80, 89 =1560sq
              _______
              + =6660

              Yep! And, isn't that INCREDIBLE! You could try adding in the area from the
              20,99,101 (990sq -it's like number mirror- 6660/990) triple, for fun, that'd
              give ya 7,650 =2x2x2x5x5x13.

              Did you notice that the 5,12,13 (many of these triples turn out to be quite
              interesting BUT this is my favorite triple thus far) is equal in both its
              quanity/perimeter value (30) and its area (30sq) -isn't that great! Also,
              about this triple (5,12,13) when I tried to connect the 14 triangles
              together this was the only triple where all three of its numbers
              interconnected to other triples -see attachment- All together I was able to
              form two pieces with 7 of those triangles -basically fitting 5 into one
              (looks like a bird) and 2 into another (looks like a crystal). Here's
              another interesting note -see attachment- of all 42 of these numbers listed
              in/as triples only 5 repeat (or, are used more than once in the grid) those
              numbers are 5,12,13,65,85 -ooh, and, 5+12+13+65+85=180 X 2 (being that they
              are repeated) =360!

              This grid is just so beautiful!!

              (Side note) I was looking at the 22,120,122 triple and found the 5280
              -number of feet in a mile. That's interesting too!

              -AND- Thanks for that web address -I like it a lot!

              Have a BEAUTIFUL day,

              Terrie





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            • Terrie Halprin
              Hi Y all, When I look at these triples I can barely believe all the disertation that s happening within them. Man! I m gonna read these triples like a really
              Message 6 of 6 , Aug 2, 2001
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                Hi Y'all,

                When I look at these triples I can barely believe all the disertation that's
                happening within them. Man! I'm gonna read these triples like a really
                good book -one that one just simply comes to adore!

                So, let's just start from the beginning and try and go cover to cover.
                First triples first and so on: (I've put perimeter sums under column )

                MICRO MACRO MACRO (-micro)

                1. 12 | 6sq | 12 24 |
                2. 30 | 30sq | 30 36 |
                3. 56 | 60sq | 24 60 |
                4. 40 | 84sq | 56 48 |
                5. 90 | 180sq | 40 112 |
                6. 132 | --+=360 36 60 |
                --+=360

                So, the first five triples, as area, =360 AND the first 6 triples, as
                perimeter, =360! Oh, man! That the very first subject these darlings would
                example for us is 360/the circle is just so very BEAUTIFUL -and- yeah, it
                does so in very precise geometric terms -check it out- (total area sums) 6,
                (+30) =36, (+84) =120, (+60) =180, (+180) =360 -that's 1/60th, 1/10th,
                1/4th, 1/2 and 1 !

                7. 84 * 90 90 |
                8. 182 60 80 |
                9. 144 | 132 72 |
                10 70 | 48 * 180
                11 126 | 84 120
                12 154 | 182 84
                13 198 | 112 168
                14 208 | 60 96
                15 176 90 120
                16 234 80 234

                Neil, when you included that micro triple (12,35,37) into the macro grid MAN
                it was such a brilliant/intuitive stroke -the god(s) must've been whispering
                in your ear I think cuz there's something real interesting about this for us
                to look to understand. I'm taking a look! You'll note that in the macro
                (-micro) grid of perimeter sums (right) I've marked off 9 triple sums with
                "|", these sums are 24,36 (24+36=60),60,48,112,60,90,80,72 +=582 -watch this
                get interesting quick because in that micro triples grid (with
                perimeter/area sums) we've already sort of seen the first 5 and 6 placements
                fashioned off -so, really, in a sense, the next available triples sum to set
                into some sort of perimeter/area (to look at more thoroughly) context is the
                6/7 placements -and- oddly enough, if we take those 9triples perimeter sums
                from that macro (-micro) and add on the very next perimeter sum (84 which I
                noted by *) from the micro grid (12,35,37 =84) YEP it's the 666. This is
                interesting reflection process/an interesting idea that there's natal
                utility to interacting between macro/micro grids and/or context. So, I take
                a look, I look for the 1080 -sure enough, it's in here. You'll note in the
                micro grid that I've marked off 6 triples sums by "|"
                (144,70,126,154,198,208,176,162 +=900) AND "*" the very next available
                triple sum (back and forth, bach and forth) from the macro (minus micro)
                grid 18,80,82 (=180) AND 900+180 =1080! Isn't that beautiful -there's no
                telling what's next -the great year, the celestrial marriage -who knows-
                we'll just have to do the math AND try our best to follow through with some
                of these emerging patternings BUT oh, man, it's just so fascinating to
                consider just how one might watch this Loveely operating. You'll note the
                182 in the micro grid is not as yet accounted for (actually, there are 3
                remaining perimeter sums in the micro grid which are unaccounted for) -This
                triples sum, such a helper(!) in deducting 1080 -cuz- when I was adding up
                remaining micro perimeter sums, (from the 8th sum on up to the 16) of
                course, I got to a 1082 and thought -wait just a minute here, and, sure
                enough, there it was, a 180 in the macro (-micro) grid just hanging out,
                waiting to be next and entirely capable of creating the exact. So, this sum
                (182), it shows us a kindness -AND- it'd be such a great pleasure to place
                it into good context. Ooh, I think that's interesting too that the very
                next available sums in either the micro or the macro grids would be such
                relatively close (182 and 180) digits.

                Number placements arrangments thus far.... Micro: first 5 (area) =360,
                first 6 (perimeter) =360 -Macro: first 9 perimeter sums (+7th micro
                perimeter sum) =666, then, -next- the 10th macro perimeter sum (+ 9-14 micro
                perimeters sums) =1080. It's all so interesting! Yep, I'm liking (getting
                into) the idea of working these grids.

                Neil, Mike, I like how the micro triples grid can be expanded to the
                16/square, really, from, maybe, as far back as 5th/6th or the 7th (perimeter
                under 100) but, I wonder if that geometry might be able to be expanded
                anymore, maybe, to be as big as the 20 which c/would be kind of fun cuz, oh,
                yeah, love that square(!) -BUT- we could see about fashioning a box for a
                box. Also, I wonder about the 52 too cuz it'd be nice to get this digit to
                the 60. Oh, yeah, baby(!), 52 weeks is nice/current and the 7 split in the
                14 is/c/would be real nice/easy to work with -BUT- oh, man, if we could get
                on to the 60 -6 days in a week- 365 divided by 6 =60.8333 weeks in a year
                -and/or- exactly 61 on every leap year! That would be BEAUTIFUL!


                Have a BEAUTIFUL day,

                Terrie

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