[math-fun] Dissection problem
1. Dissect a triangle of your choice into n > 1 triangles with all 3n edge lengths distinct. 2. Same question for a quadrilateral of your choice.
Any interior edge is common to two small triangles, so all interior sides must be equal in pairs? WFL On 6/9/16, David Wilson <davidwwilson@comcast.net> wrote:
1. Dissect a triangle of your choice into n > 1 triangles with all 3n edge lengths distinct. 2. Same question for a quadrilateral of your choice.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
On Jun 9, 2016, at 10:34 AM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Any interior edge is common to two small triangles, so all interior sides must be equal in pairs? WFL
True, the dissection/tiling cannot be edge-to-edge. But consider a triangle, and mark one point on each of its edges, always within the first half in a clockwise sense. Joining vertices to marked points on opposite edges will form an internal triangle — that is one of the triangles of the dissection. I’ll leave it to you to find the other three. -Veit
It can't be edge-to-edge *anywhere. *I am not seeing how to do this at all. On Thu, Jun 9, 2016 at 10:58 AM, Veit Elser <ve10@cornell.edu> wrote:
On Jun 9, 2016, at 10:34 AM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Any interior edge is common to two small triangles, so all interior sides must be equal in pairs? WFL
True, the dissection/tiling cannot be edge-to-edge. But consider a triangle, and mark one point on each of its edges, always within the first half in a clockwise sense. Joining vertices to marked points on opposite edges will form an internal triangle — that is one of the triangles of the dissection. I’ll leave it to you to find the other three.
-Veit _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
And that's why it is a puzzle. On Thu, Jun 9, 2016 at 8:13 AM, Allan Wechsler <acwacw@gmail.com> wrote:
It can't be edge-to-edge *anywhere. *I am not seeing how to do this at all.
On Thu, Jun 9, 2016 at 10:58 AM, Veit Elser <ve10@cornell.edu> wrote:
On Jun 9, 2016, at 10:34 AM, Fred Lunnon <fred.lunnon@gmail.com>
wrote:
Any interior edge is common to two small triangles, so all interior sides must be equal in pairs? WFL
True, the dissection/tiling cannot be edge-to-edge. But consider a triangle, and mark one point on each of its edges, always within the first half in a clockwise sense. Joining vertices to marked points on opposite edges will form an internal triangle — that is one of the triangles of the dissection. I’ll leave it to you to find the other three.
-Veit _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- -- http://cube20.org/ -- [ <http://golly.sf.net/>Golly link suppressed; ask me why] --
Have a look at my last post in FB: https://www.facebook.com/zak.seidov/allactivity?privacy_source=activity_log&... Zak
Четверг, 9 июня 2016, 18:45 +03:00 от Tom Rokicki <rokicki@gmail.com>:
And that's why it is a puzzle.
On Thu, Jun 9, 2016 at 8:13 AM, Allan Wechsler < acwacw@gmail.com > wrote:
It can't be edge-to-edge *anywhere. *I am not seeing how to do this at all.
On Thu, Jun 9, 2016 at 10:58 AM, Veit Elser < ve10@cornell.edu > wrote:
On Jun 9, 2016, at 10:34 AM, Fred Lunnon < fred.lunnon@gmail.com >
wrote:
Any interior edge is common to two small triangles, so all interior sides must be equal in pairs? WFL
True, the dissection/tiling cannot be edge-to-edge. But consider a triangle, and mark one point on each of its edges, always within the first half in a clockwise sense. Joining vertices to marked points on opposite edges will form an internal triangle — that is one of the triangles of the dissection. I’ll leave it to you to find the other three.
-Veit _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- -- http://cube20.org/ -- [ < http://golly.sf.net/ >Golly link suppressed; ask me why] -- _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
A 16-19-21 triangle can be divided into 5-7-8, 9-15-21, 6-14-16, and 7-13-19, with the 7 edges not touching. I posted four solutions at http://math.stackexchange.com/questions/1819928/triangle-dissection-no-share... On Thu, Jun 9, 2016 at 11:03 AM, Zak Seidov <math-fun@mailman.xmission.com> wrote:
Have a look at my last post in FB:
https://www.facebook.com/zak.seidov/allactivity?privacy_source=activity_log&...
Zak
Четверг, 9 июня 2016, 18:45 +03:00 от Tom Rokicki <rokicki@gmail.com>:
And that's why it is a puzzle.
On Thu, Jun 9, 2016 at 8:13 AM, Allan Wechsler < acwacw@gmail.com > wrote:
It can't be edge-to-edge *anywhere. *I am not seeing how to do this at all.
On Thu, Jun 9, 2016 at 10:58 AM, Veit Elser < ve10@cornell.edu > wrote:
On Jun 9, 2016, at 10:34 AM, Fred Lunnon < fred.lunnon@gmail.com >
wrote:
Any interior edge is common to two small triangles, so all interior sides must be equal in pairs? WFL
True, the dissection/tiling cannot be edge-to-edge. But consider a triangle, and mark one point on each of its edges, always within the first half in a clockwise sense. Joining vertices to marked points on opposite edges will form an internal triangle — that is one of the triangles of the dissection. I’ll leave it to you to find the other three.
-Veit _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- -- http://cube20.org/ -- [ < http://golly.sf.net/ >Golly link suppressed; ask me why] -- _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
The second solution looks wrong; it appears there are two small triangles with a shared edge. What do you mean by "no subtriangles are allowed"? On Thu, Jun 9, 2016 at 9:57 AM, Ed Pegg Jr <ed@mathpuzzle.com> wrote:
A 16-19-21 triangle can be divided into 5-7-8, 9-15-21, 6-14-16, and 7-13-19, with the 7 edges not touching.
I posted four solutions at
http://math.stackexchange.com/questions/1819928/triangle-dissection-no-share...
On Thu, Jun 9, 2016 at 11:03 AM, Zak Seidov <math-fun@mailman.xmission.com
wrote:
Have a look at my last post in FB:
https://www.facebook.com/zak.seidov/allactivity?privacy_source=activity_log&...
Zak
Четверг, 9 июня 2016, 18:45 +03:00 от Tom Rokicki <rokicki@gmail.com>:
And that's why it is a puzzle.
On Thu, Jun 9, 2016 at 8:13 AM, Allan Wechsler < acwacw@gmail.com > wrote:
It can't be edge-to-edge *anywhere. *I am not seeing how to do this at all.
On Thu, Jun 9, 2016 at 10:58 AM, Veit Elser < ve10@cornell.edu >
wrote:
On Jun 9, 2016, at 10:34 AM, Fred Lunnon < fred.lunnon@gmail.com
wrote:
Any interior edge is common to two small triangles, so all interior sides must be equal in pairs? WFL
True, the dissection/tiling cannot be edge-to-edge. But consider a triangle, and mark one point on each of its edges, always within the first half in a clockwise sense. Joining vertices to marked points on opposite edges will form an internal triangle — that is one of the triangles of the dissection. I’ll leave it to you to find the other three.
-Veit _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- -- http://cube20.org/ -- [ < http://golly.sf.net/ >Golly link suppressed; ask me why] -- _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- -- http://cube20.org/ -- [ <http://golly.sf.net/>Golly link suppressed; ask me why] --
I stand well bamboozled! WFL On 6/9/16, Tom Rokicki <rokicki@gmail.com> wrote:
The second solution looks wrong; it appears there are two small triangles with a shared edge.
What do you mean by "no subtriangles are allowed"?
On Thu, Jun 9, 2016 at 9:57 AM, Ed Pegg Jr <ed@mathpuzzle.com> wrote:
A 16-19-21 triangle can be divided into 5-7-8, 9-15-21, 6-14-16, and 7-13-19, with the 7 edges not touching.
I posted four solutions at
http://math.stackexchange.com/questions/1819928/triangle-dissection-no-share...
On Thu, Jun 9, 2016 at 11:03 AM, Zak Seidov <math-fun@mailman.xmission.com
wrote:
Have a look at my last post in FB:
https://www.facebook.com/zak.seidov/allactivity?privacy_source=activity_log&...
Zak
Четверг, 9 июня 2016, 18:45 +03:00 от Tom Rokicki <rokicki@gmail.com>:
And that's why it is a puzzle.
On Thu, Jun 9, 2016 at 8:13 AM, Allan Wechsler < acwacw@gmail.com > wrote:
It can't be edge-to-edge *anywhere. *I am not seeing how to do this at all.
On Thu, Jun 9, 2016 at 10:58 AM, Veit Elser < ve10@cornell.edu >
wrote:
> On Jun 9, 2016, at 10:34 AM, Fred Lunnon < fred.lunnon@gmail.com
wrote:
> > Any interior edge is common to two small triangles, > so all interior sides must be equal in pairs? WFL
True, the dissection/tiling cannot be edge-to-edge. But consider a triangle, and mark one point on each of its edges, always within the first half in a clockwise sense. Joining vertices to marked points on opposite edges will form an internal triangle — that is one of the triangles of the dissection. I’ll leave it to you to find the other three.
-Veit _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- -- http://cube20.org/ -- [ < http://golly.sf.net/ >Golly link suppressed; ask me why] -- _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- -- http://cube20.org/ -- [ <http://golly.sf.net/>Golly link suppressed; ask me why] -- _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
I've managed to solve both. On Thu, Jun 9, 2016 at 2:42 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
I stand well bamboozled! WFL
On 6/9/16, Tom Rokicki <rokicki@gmail.com> wrote:
The second solution looks wrong; it appears there are two small triangles with a shared edge.
What do you mean by "no subtriangles are allowed"?
On Thu, Jun 9, 2016 at 9:57 AM, Ed Pegg Jr <ed@mathpuzzle.com> wrote:
A 16-19-21 triangle can be divided into 5-7-8, 9-15-21, 6-14-16, and 7-13-19, with the 7 edges not touching.
I posted four solutions at
http://math.stackexchange.com/questions/1819928/triangle-dissection-no-share...
On Thu, Jun 9, 2016 at 11:03 AM, Zak Seidov <
math-fun@mailman.xmission.com
wrote:
Have a look at my last post in FB:
https://www.facebook.com/zak.seidov/allactivity?privacy_source=activity_log&...
Zak
Четверг, 9 июня 2016, 18:45 +03:00 от Tom Rokicki <
rokicki@gmail.com>:
And that's why it is a puzzle.
On Thu, Jun 9, 2016 at 8:13 AM, Allan Wechsler < acwacw@gmail.com >
wrote:
It can't be edge-to-edge *anywhere. *I am not seeing how to do this at
all.
On Thu, Jun 9, 2016 at 10:58 AM, Veit Elser < ve10@cornell.edu >
wrote:
> > > On Jun 9, 2016, at 10:34 AM, Fred Lunnon <
fred.lunnon@gmail.com
wrote: > > > > Any interior edge is common to two small triangles, > > so all interior sides must be equal in pairs? WFL > > True, the dissection/tiling cannot be edge-to-edge. But consider a > triangle, and mark one point on each of its edges, always within > the first > half in a clockwise sense. Joining vertices to marked points on opposite > edges will form an internal triangle — that is one of the triangles of the > dissection. I’ll leave it to you to find the other three. > > -Veit > _______________________________________________ > math-fun mailing list > math-fun@mailman.xmission.com > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- -- http://cube20.org/ -- [ < http://golly.sf.net/ >Golly link suppressed; ask me why] -- _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- -- http://cube20.org/ -- [ <http://golly.sf.net/>Golly link suppressed; ask me why] -- _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- -- http://cube20.org/ -- [ <http://golly.sf.net/>Golly link suppressed; ask me why] --
I'd really be interested in the quadrilateral dissection, since that, I believe, would imply a dissection of any polygon.
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Tom Rokicki Sent: Thursday, June 09, 2016 5:46 PM To: math-fun Subject: Re: [math-fun] Dissection problem
I've managed to solve both.
On Thu, Jun 9, 2016 at 2:42 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
I stand well bamboozled! WFL
On 6/9/16, Tom Rokicki <rokicki@gmail.com> wrote:
The second solution looks wrong; it appears there are two small triangles with a shared edge.
What do you mean by "no subtriangles are allowed"?
On Thu, Jun 9, 2016 at 9:57 AM, Ed Pegg Jr <ed@mathpuzzle.com> wrote:
A 16-19-21 triangle can be divided into 5-7-8, 9-15-21, 6-14-16, and 7-13-19, with the 7 edges not touching.
I posted four solutions at
http://math.stackexchange.com/questions/1819928/triangle-dissection-no -shared-edges
On Thu, Jun 9, 2016 at 11:03 AM, Zak Seidov <
math-fun@mailman.xmission.com
wrote:
Have a look at my last post in FB:
https://www.facebook.com/zak.seidov/allactivity?privacy_source=activit y_log&log_filter=cluster_11
Zak
Четверг, 9 июня 2016, 18:45 +03:00 от Tom Rokicki <
rokicki@gmail.com>:
And that's why it is a puzzle.
On Thu, Jun 9, 2016 at 8:13 AM, Allan Wechsler < acwacw@gmail.com >
wrote:
> It can't be edge-to-edge *anywhere. *I am not seeing how to do > this at
all.
> > > On Thu, Jun 9, 2016 at 10:58 AM, Veit Elser < ve10@cornell.edu > > wrote: > > > > > > On Jun 9, 2016, at 10:34 AM, Fred Lunnon < fred.lunnon@gmail.com
> wrote: > > > > > > Any interior edge is common to two small triangles, > > > so all interior sides must be equal in pairs? WFL > > > > True, the dissection/tiling cannot be edge-to-edge. But > > consider a > > triangle, and mark one point on each of its edges, always > > within the > first > > half in a clockwise sense. Joining vertices to marked points > > on opposite > > edges will form an internal triangle — that is one of the triangles of > the > > dissection. I’ll leave it to you to find the other three. > > > > -Veit > > _______________________________________________ > > math-fun mailing list > > math-fun@mailman.xmission.com > > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-f > > un > > > _______________________________________________ > math-fun mailing list > math-fun@mailman.xmission.com > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >
-- -- http://cube20.org/ -- [ < http://golly.sf.net/ >Golly link suppressed; ask me why] -- _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- -- http://cube20.org/ -- [ <http://golly.sf.net/>Golly link suppressed; ask me why] -- _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- -- http://cube20.org/ -- [ <http://golly.sf.net/>Golly link suppressed; ask me why] -- _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Or rather, the convex quadrilateral dissection.
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of David Wilson Sent: Thursday, June 09, 2016 6:24 PM To: 'math-fun' Subject: Re: [math-fun] Dissection problem
I'd really be interested in the quadrilateral dissection, since that, I believe, would imply a dissection of any polygon.
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Tom Rokicki Sent: Thursday, June 09, 2016 5:46 PM To: math-fun Subject: Re: [math-fun] Dissection problem
I've managed to solve both.
On Thu, Jun 9, 2016 at 2:42 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
I stand well bamboozled! WFL
On 6/9/16, Tom Rokicki <rokicki@gmail.com> wrote:
The second solution looks wrong; it appears there are two small triangles with a shared edge.
What do you mean by "no subtriangles are allowed"?
On Thu, Jun 9, 2016 at 9:57 AM, Ed Pegg Jr <ed@mathpuzzle.com> wrote:
A 16-19-21 triangle can be divided into 5-7-8, 9-15-21, 6-14-16, and 7-13-19, with the 7 edges not touching.
I posted four solutions at
http://math.stackexchange.com/questions/1819928/triangle-dissection- no -shared-edges
On Thu, Jun 9, 2016 at 11:03 AM, Zak Seidov <
math-fun@mailman.xmission.com
wrote:
Have a look at my last post in FB:
https://www.facebook.com/zak.seidov/allactivity?privacy_source=activ it y_log&log_filter=cluster_11
Zak
>Четверг, 9 июня 2016, 18:45 +03:00 от Tom Rokicki <
rokicki@gmail.com>:
> >And that's why it is a puzzle. > >On Thu, Jun 9, 2016 at 8:13 AM, Allan Wechsler < >acwacw@gmail.com > wrote: > >> It can't be edge-to-edge *anywhere. *I am not seeing how to >> do this at all. >> >> >> On Thu, Jun 9, 2016 at 10:58 AM, Veit Elser < >> ve10@cornell.edu >> > wrote: >> >> > >> > > On Jun 9, 2016, at 10:34 AM, Fred Lunnon < fred.lunnon@gmail.com
>> wrote: >> > > >> > > Any interior edge is common to two small triangles, >> > > so all interior sides must be equal in pairs? WFL >> > >> > True, the dissection/tiling cannot be edge-to-edge. But >> > consider a >> > triangle, and mark one point on each of its edges, always >> > within the >> first >> > half in a clockwise sense. Joining vertices to marked >> > points on opposite >> > edges will form an internal triangle — that is one of the triangles of >> the >> > dissection. I’ll leave it to you to find the other three. >> > >> > -Veit >> > _______________________________________________ >> > math-fun mailing list >> > math-fun@mailman.xmission.com >> > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math >> > -f >> > un >> > >> _______________________________________________ >> math-fun mailing list >> math-fun@mailman.xmission.com >> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-f >> un >> > > > >-- >-- http://cube20.org/ -- [ < http://golly.sf.net/ >Golly >link suppressed; >ask me why] -- >_______________________________________________ >math-fun mailing list >math-fun@mailman.xmission.com >https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- -- http://cube20.org/ -- [ <http://golly.sf.net/>Golly link suppressed; ask me why] -- _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- -- http://cube20.org/ -- [ <http://golly.sf.net/>Golly link suppressed; ask me why] -- _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
I fixed the errors at http://math.stackexchange.com/questions/1819928/triangle-dissection-no-share... and found a new solution. http://i.imgur.com/F00ldIl.gif That has 19 internal triangles. Has anyone found new solutions with 7 to 18 triangles? Ed Pegg Jr On Thu, Jun 9, 2016 at 7:56 PM, David Wilson <davidwwilson@comcast.net> wrote:
Or rather, the convex quadrilateral dissection.
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of David Wilson Sent: Thursday, June 09, 2016 6:24 PM To: 'math-fun' Subject: Re: [math-fun] Dissection problem
I'd really be interested in the quadrilateral dissection, since that, I believe, would imply a dissection of any polygon.
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Tom Rokicki Sent: Thursday, June 09, 2016 5:46 PM To: math-fun Subject: Re: [math-fun] Dissection problem
I've managed to solve both.
On Thu, Jun 9, 2016 at 2:42 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
I stand well bamboozled! WFL
On 6/9/16, Tom Rokicki <rokicki@gmail.com> wrote:
The second solution looks wrong; it appears there are two small triangles with a shared edge.
What do you mean by "no subtriangles are allowed"?
On Thu, Jun 9, 2016 at 9:57 AM, Ed Pegg Jr <ed@mathpuzzle.com> wrote:
A 16-19-21 triangle can be divided into 5-7-8, 9-15-21, 6-14-16, and 7-13-19, with the 7 edges not touching.
I posted four solutions at
http://math.stackexchange.com/questions/1819928/triangle-dissection- no -shared-edges
On Thu, Jun 9, 2016 at 11:03 AM, Zak Seidov <
math-fun@mailman.xmission.com
> wrote:
> Have a look at my last post in FB: > > >
https://www.facebook.com/zak.seidov/allactivity?privacy_source=activ it y_log&log_filter=cluster_11
> > > Zak > > > >Четверг, 9 июня 2016, 18:45 +03:00 от Tom Rokicki < rokicki@gmail.com>: > > > >And that's why it is a puzzle. > > > >On Thu, Jun 9, 2016 at 8:13 AM, Allan Wechsler < > >acwacw@gmail.com > > wrote: > > > >> It can't be edge-to-edge *anywhere. *I am not seeing how to > >> do this at > all. > >> > >> > >> On Thu, Jun 9, 2016 at 10:58 AM, Veit Elser < > >> ve10@cornell.edu > >> > wrote: > >> > >> > > >> > > On Jun 9, 2016, at 10:34 AM, Fred Lunnon < fred.lunnon@gmail.com > > >> wrote: > >> > > > >> > > Any interior edge is common to two small triangles, > >> > > so all interior sides must be equal in pairs? WFL > >> > > >> > True, the dissection/tiling cannot be edge-to-edge. But > >> > consider a > >> > triangle, and mark one point on each of its edges, always > >> > within the > >> first > >> > half in a clockwise sense. Joining vertices to marked > >> > points on > opposite > >> > edges will form an internal triangle — that is one of the triangles of > >> the > >> > dissection. I’ll leave it to you to find the other three. > >> > > >> > -Veit > >> > _______________________________________________ > >> > math-fun mailing list > >> > math-fun@mailman.xmission.com > >> > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math > >> > -f > >> > un > >> > > >> _______________________________________________ > >> math-fun mailing list > >> math-fun@mailman.xmission.com > >> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-f > >> un > >> > > > > > > > >-- > >-- http://cube20.org/ -- [ < http://golly.sf.net/ >Golly > >link > suppressed; > >ask me why] -- > >_______________________________________________ > >math-fun mailing list > >math-fun@mailman.xmission.com > >https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > _______________________________________________ > math-fun mailing list > math-fun@mailman.xmission.com > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- -- http://cube20.org/ -- [ <http://golly.sf.net/>Golly link suppressed; ask me why] -- _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- -- http://cube20.org/ -- [ <http://golly.sf.net/>Golly link suppressed; ask me why] -- _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
No. Worked out a proof many years ago. Also: dissect a pentagon into non edge to edge pentagons (minimum=11),pentagon into convex pentagons (impossible), hexagon into hexagons (impossible), triangle into convex quads (minimum = 9). On Sunday, June 12, 2016, David Wilson <davidwwilson@comcast.net> wrote:
Is there any dissection for a convex quadrilateral?
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Gee, don't I get any credit?
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Ed Pegg Jr Sent: Thursday, June 09, 2016 12:58 PM To: Zak Seidov; math-fun Subject: Re: [math-fun] Dissection problem
A 16-19-21 triangle can be divided into 5-7-8, 9-15-21, 6-14-16, and 7-13-19, with the 7 edges not touching.
I posted four solutions at http://math.stackexchange.com/questions/1819928/triangle-dissection-no- shared-edges
On Thu, Jun 9, 2016 at 11:03 AM, Zak Seidov <math- fun@mailman.xmission.com> wrote:
Have a look at my last post in FB:
https://www.facebook.com/zak.seidov/allactivity?privacy_source=activit y_log&log_filter=cluster_11
Zak
Четверг, 9 июня 2016, 18:45 +03:00 от Tom Rokicki <rokicki@gmail.com>:
And that's why it is a puzzle.
On Thu, Jun 9, 2016 at 8:13 AM, Allan Wechsler < acwacw@gmail.com > wrote:
It can't be edge-to-edge *anywhere. *I am not seeing how to do this at all.
On Thu, Jun 9, 2016 at 10:58 AM, Veit Elser < ve10@cornell.edu > wrote:
On Jun 9, 2016, at 10:34 AM, Fred Lunnon < fred.lunnon@gmail.com >
wrote:
Any interior edge is common to two small triangles, so all interior sides must be equal in pairs? WFL
True, the dissection/tiling cannot be edge-to-edge. But consider a triangle, and mark one point on each of its edges, always within the first half in a clockwise sense. Joining vertices to marked points on opposite edges will form an internal triangle — that is one of the triangles of the dissection. I’ll leave it to you to find the other three.
-Veit _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- -- http://cube20.org/ -- [ < http://golly.sf.net/ >Golly link suppressed; ask me why] -- _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (8)
-
Allan Wechsler -
David Wilson -
Ed Pegg Jr -
Fred Lunnon -
Scott Kim -
Tom Rokicki -
Veit Elser -
Zak Seidov