Re: [math-fun] Fano Plane puzzle
The Sierra Club has you beaten. They've quoted the volume of nuclear waste in cubic liters. -- Gene Why do only bad things come in cubic liters? I saw a GreenPeace flyer celebrating the sabotage of a drainpipe dumping "thousands of cubic liters of pharmaceutical waste" into the ocean. Unfortunately, the photo showed only three-dimensional damage, leaving hundreds of square gallons unspilled, resulting in attacks by fifteen pound purple jellyfish: http://sanfrancisco.cbslocal.com/2015/06/12/big-purple-sea-slugs-wash-ashore... --rwg From: rwg <rwg@sdf.org> To: math-fun <math-fun@mailman.xmission.com> Sent: Wednesday, June 24, 2015 6:28 PM Subject: Re: [math-fun] Fano Plane puzzle On 2015-06-24 17:07, Dan Asimov wrote: On Jun 24, 2015, at 3:22 PM, David Wilson <davidwwilson@comcast.net> wrote: Sadly, it would difficult to make an office toy in R^6. Not at all. In R^6 they have even better machine shops than we do. —Dan And the wealth to operate them. Like I was just telling the kids, On Wed, Jun 24, 2015 at 7:19 AM, Bill Gosper <billgosper@gmail.com> wrote: Subj: Scrooge McDuck was said to have three cubic acres of money. How much is that in square gallons? In[169]:= UnitConvert[Quantity[3, "Acres"]^3, Quantity["Gallons"]^2] Out[169]= Quantity[95610810544128000000000000000000000000000000000000000/ 765615812545937377500183749853000049, ("Gallons")^2] Oops, that's 27 cubic acres! Should have been In[170]:= UnitConvert[Quantity[3, "Acres"^3], Quantity["Gallons"]^2] Out[170]= Quantity[10623423393792000000000000000000000000000000000000000/ 765615812545937377500183749853000049, ("Gallons")^2] I hate it when ducks exaggerate. --Bill -----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com <mailto:math-fun-bounces@mailman.xmission.com>] Dan Asimov Sent: Wednesday, June 24, 2015 3:30 PM But there could be beautiful configurations for the Fano plane in higher dimensions. Maybe in 6-space, where the vertices of the unit simplex s_6 form 140 unit equilateral triangles.
I put all the details of my unit distance Heawood graph at http://community.wolfram.com/groups/-/m/t/518461 --Ed Pegg Jr On Thu, Jun 25, 2015 at 10:53 AM, Bill Gosper <billgosper@gmail.com> wrote:
The Sierra Club has you beaten. They've quoted the volume of nuclear waste in cubic liters.
-- Gene
Why do only bad things come in cubic liters? I saw a GreenPeace flyer celebrating the sabotage of a drainpipe dumping "thousands of cubic liters of pharmaceutical waste" into the ocean. Unfortunately, the photo showed only three-dimensional damage, leaving hundreds of square gallons unspilled, resulting in attacks by fifteen pound purple jellyfish:
http://sanfrancisco.cbslocal.com/2015/06/12/big-purple-sea-slugs-wash-ashore... --rwg
From: rwg <rwg@sdf.org> To: math-fun <math-fun@mailman.xmission.com> Sent: Wednesday, June 24, 2015 6:28 PM Subject: Re: [math-fun] Fano Plane puzzle
On 2015-06-24 17:07, Dan Asimov wrote:
On Jun 24, 2015, at 3:22 PM, David Wilson <davidwwilson@comcast.net> wrote:
Sadly, it would difficult to make an office toy in R^6.
Not at all. In R^6 they have even better machine shops than we do.
—Dan
And the wealth to operate them. Like I was just telling the kids,
On Wed, Jun 24, 2015 at 7:19 AM, Bill Gosper <billgosper@gmail.com> wrote: Subj: Scrooge McDuck
was said to have three cubic acres of money. How much is that in square gallons?
In[169]:= UnitConvert[Quantity[3, "Acres"]^3, Quantity["Gallons"]^2]
Out[169]= Quantity[95610810544128000000000000000000000000000000000000000/ 765615812545937377500183749853000049, ("Gallons")^2]
Oops, that's 27 cubic acres! Should have been
In[170]:= UnitConvert[Quantity[3, "Acres"^3], Quantity["Gallons"]^2]
Out[170]= Quantity[10623423393792000000000000000000000000000000000000000/ 765615812545937377500183749853000049, ("Gallons")^2]
I hate it when ducks exaggerate. --Bill
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com <mailto:math-fun-bounces@mailman.xmission.com>] Dan Asimov Sent: Wednesday, June 24, 2015 3:30 PM
But there could be beautiful configurations for the Fano plane in higher dimensions. Maybe in 6-space, where the vertices of the unit simplex s_6 form 140 unit equilateral triangles. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
I took a look at Ed Pegg Jr's exact algebraic unit-length embedding of the Heawood graph at http://community.wolfram.com/groups/-/m/t/518461 Expressed in terms of the single real root z of a degree-21 polynomial with 9-digit coefficients, it surely is a considerable improvement over the one Dan reviewed earlier. There are minor difficulties with the webpage definition. Although his list "pts" of vertices specifies their coordinates components correctly, they appear in an order disobeying his bipartite classification via index parity: this may be corrected by reversing signs of the final 6 x-components. Furthermore his ordering ignores any 1-2-4 rule facilitating correspondence between points and circles. When modifying his Mathematica script for Maple I was therefore eventually obliged to radically unscramble them. Plotted in the form of a Fano plane, the embedding appears as https://www.dropbox.com/s/3qv94ot8t2jw58b/fano_pegg.gif?dl=0 Everything looks tidily spaced apart ... hang on while I polish my specs ... 'Ullo, 'ullo, 'ullo, wot's this lurking down near (-1.4, -0.7) then? Looks uncommonly like another three-circle intersection ?! Indeed there is an eighth triple point precisely at (-1-x, y) , where (x, y) denotes the Fano point near (0.4, -0.7) ; and dually, an eighth unit circle centred at (1+x, y) meeting all three Fano points with x > 0 & y < 0 ! Won't please DWW then, but might appeal to RWG instead. Fred Lunnon # Maple data: z := 'z': zpol := - 153027 + 1353114*z - 1986024*z^2 + 2763312*z^3 + 10497808*z^4 + 31063072*z^5 - 8457728*z^6 - 40590336*z^7 + 44468736*z^8 + 164975616*z^9 + 119705600*z^10 - 494067712*z^11 - 763912192*z^12 + 414924800*z^13 + 1154613248*z^14 + 48496640*z^15 - 864485376*z^16 - 376307712*z^17 + 253231104*z^18 + 271581184*z^19 + 89128960*z^20 + 10485760*z^21; z := fsolve(zpol, z, real); # 0.1304718735 --- single real root! # Centres and points are alternate Heawood vertices; # if (x,y) is a point then (-x,y) is a centre epjpnt := [ [z, 1/2], # 0 [-1/2, 1/2*(-1+sqrt(3-4*z-4*z^2))], # 1 [1/2, -(1/2)*(-1+sqrt(3-4*z-4*z^2))], # 2 [-sqrt(3-4*z^2)/(2*sqrt(1+4*z^2)), -((z*sqrt(3-4*z^2))/sqrt(1+4*z^2))], # 3 [1/2,-((z*sqrt(3-4*z^2))/sqrt(1+4*z^2))-sqrt((8*z^2+sqrt(3+8*z^2-16*z^4))/(2+8*z^2))], # 4 [-z, -1/2], # 5 [(1/2)*(-1+sqrt(3-4*z*(1+z)))*sqrt((7+8*sqrt(3-4*z*(1+z)) -8*z*(-1+z*(1+2*z*(2+z))))/(13+8*z*(1+z)*(-3+2*z*(1+z)))), -(1/2)*sqrt((7+8*sqrt(3-4*z*(1+z))-8*z*(-1+z*(1+2*z*(2+z))))/(13+8*z*(1+z)*(-3+2*z*(1+z))))], # ?? NULL]: epjpnt := evalf(epjpnt); # epjpnt := [ [.1304718735, .5000000000], [-.5000000000, .2762120950], [.5000000000, -.2762120950], [-.8284016255, -.2161662241], [.5000000000, -1.160704403], [-.1304718735, -.5000000000], [.3968937166, -.7184582495] ]: # eighth_point := [-1.396893718, -.7184582435]:
Really, all I want is a stabile hanging in my office consisting of circular wire rings of equal radius (in rainbow colors) welded together with Fano connectivity and otherwise not touching one another. It's not like I'm hard to please. You would think I had asked for circular Borromean rings. Sheesh.
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Fred Lunnon Sent: Saturday, June 27, 2015 6:33 PM To: math-fun Cc: Ed Pegg Jr Subject: Re: [math-fun] Fano Plane puzzle
[Stuff I took out]
Won't please DWW then, but might appeal to RWG instead.
Fred Lunnon
On 6/28/15, David Wilson <davidwwilson@comcast.net> wrote:
Really, all I want is a stabile hanging in my office consisting of circular wire rings of equal radius (in rainbow colors) welded together with Fano connectivity and otherwise not touching one another.
It's not like I'm hard to please. You would think I had asked for circular Borromean rings. Sheesh.
I had fantasised off-and-on about 7 unit hoops connected via some sort of 3-way universal sliding joints, assembed into a configuration already known (EPJ, FWH etc.), then perturbed manually. The problem is these pesky (2-ring) collisions, which almost certainly immobilise the thing straight away, unless the hoops have been a priori interlocked in some hard-to-predict fashion. I have also speculated (at greater length) about a geometry viewer permitting construction of mechanisms with similar specified constraints, followed by arbitrary user-driven perturbations within the remaining freedom. There may well be some physics/engineering simulation package out there suitable: I have never seriously investigated. With one of them you could have your "stabile" mobile on your screensaver as well! WFL
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Fred Lunnon Sent: Saturday, June 27, 2015 6:33 PM To: math-fun Cc: Ed Pegg Jr Subject: Re: [math-fun] Fano Plane puzzle
[Stuff I took out]
Won't please DWW then, but might appeal to RWG instead.
Fred Lunnon
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
On 6/28/15, David Wilson <davidwwilson@comcast.net> wrote:
Really, all I want is a stabile hanging in my office consisting of circular wire rings of equal radius (in rainbow colors) welded together with Fano connectivity and otherwise not touching one another.
It's not like I'm hard to please. You would think I had asked for circular Borromean rings. Sheesh.
Right then --- here is an elegant embedding exactly as requested (on numerous occasions) by DWW for a Fano plane composed of points and circles. It has 3-fold symmetry about axis line x = y = z , without extraneous collisions or near-misses. Four points lie at alternate corners of a scaled unit cube; the other three at equal displacements along axis lines. Approximate values for the points and centres are as follows; as before, a unit circle with centre index i (mod 7) meets points indexed i, i+1, i+3 . q := 0.5912088662; p := 1.224744871; points := [ [q,-q,-q], [-q,q,-q], [-q,-q,q], [0,p,0], [0,0,p], [q,q,q], [p,0,0] ]; u := 0.3203726673; v := 0.2819082667; w := 0.4082482903; centres := [ [u,u,-v], [-v,u,u], [-u,u,v], [w,w,w], [v,-u,u], [u,v,-u], [u,-v,u] ]; I have a proof of correctness, and exact algebraic equations for the point components, though not the centres: 2*p^2 - 3 = 0 ; 36*q^4 - 4*q^2 - 3 = 0 ; Available on request is a Maple worksheet showing a (rather feeble) 3-D manipulable graphic, which ate up an order of magnitude more coding effort than finding the configuration in the first place! Fred Lunnon
A couple of (considerably less feeble) views of solid Fano posted at https://www.dropbox.com/s/i4l1szci7o0ww7r/fano_solid0.gif?dl=0 https://www.dropbox.com/s/vpgp26kcr1ttsgk/fano_solid1.gif?dl=0 the first showing 3-fold symmetry about cube diagonal. WFL On 6/29/15, Fred Lunnon <fred.lunnon@gmail.com> wrote:
On 6/28/15, David Wilson <davidwwilson@comcast.net> wrote:
Really, all I want is a stabile hanging in my office consisting of circular wire rings of equal radius (in rainbow colors) welded together with Fano connectivity and otherwise not touching one another.
It's not like I'm hard to please. You would think I had asked for circular Borromean rings. Sheesh.
Right then --- here is an elegant embedding exactly as requested (on numerous occasions) by DWW for a Fano plane composed of points and circles. It has 3-fold symmetry about axis line x = y = z , without extraneous collisions or near-misses. Four points lie at alternate corners of a scaled unit cube; the other three at equal displacements along axis lines.
Approximate values for the points and centres are as follows; as before, a unit circle with centre index i (mod 7) meets points indexed i, i+1, i+3 . q := 0.5912088662; p := 1.224744871; points := [ [q,-q,-q], [-q,q,-q], [-q,-q,q], [0,p,0], [0,0,p], [q,q,q], [p,0,0] ]; u := 0.3203726673; v := 0.2819082667; w := 0.4082482903; centres := [ [u,u,-v], [-v,u,u], [-u,u,v], [w,w,w], [v,-u,u], [u,v,-u], [u,-v,u] ];
I have a proof of correctness, and exact algebraic equations for the point components, though not the centres: 2*p^2 - 3 = 0 ; 36*q^4 - 4*q^2 - 3 = 0 ;
Available on request is a Maple worksheet showing a (rather feeble) 3-D manipulable graphic, which ate up an order of magnitude more coding effort than finding the configuration in the first place!
Fred Lunnon
Fred, you really came through. I just knew it had to exist. It's quite beautiful.
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Fred Lunnon Sent: Monday, June 29, 2015 9:23 AM To: math-fun Subject: Re: [math-fun] Fano Plane puzzle
A couple of (considerably less feeble) views of solid Fano posted at https://www.dropbox.com/s/i4l1szci7o0ww7r/fano_solid0.gif?dl=0 https://www.dropbox.com/s/vpgp26kcr1ttsgk/fano_solid1.gif?dl=0 the first showing 3-fold symmetry about cube diagonal.
WFL
On 6/29/15, Fred Lunnon <fred.lunnon@gmail.com> wrote:
On 6/28/15, David Wilson <davidwwilson@comcast.net> wrote:
Really, all I want is a stabile hanging in my office consisting of circular wire rings of equal radius (in rainbow colors) welded together with Fano connectivity and otherwise not touching one another.
It's not like I'm hard to please. You would think I had asked for circular Borromean rings. Sheesh.
Right then --- here is an elegant embedding exactly as requested (on numerous occasions) by DWW for a Fano plane composed of points and circles. It has 3-fold symmetry about axis line x = y = z , without extraneous collisions or near-misses. Four points lie at alternate corners of a scaled unit cube; the other three at equal displacements along axis lines.
Approximate values for the points and centres are as follows; as before, a unit circle with centre index i (mod 7) meets points indexed i, i+1, i+3 . q := 0.5912088662; p := 1.224744871; points := [ [q,-q,-q], [-q,q,-q], [-q,-q,q], [0,p,0], [0,0,p], [q,q,q], [p,0,0] ]; u := 0.3203726673; v := 0.2819082667; w := 0.4082482903; centres := [ [u,u,-v], [-v,u,u], [-u,u,v], [w,w,w], [v,-u,u], [u,v,-u], [u,-v,u] ];
I have a proof of correctness, and exact algebraic equations for the point components, though not the centres: 2*p^2 - 3 = 0 ; 36*q^4 - 4*q^2 - 3 = 0 ;
Available on request is a Maple worksheet showing a (rather feeble) 3-D manipulable graphic, which ate up an order of magnitude more coding effort than finding the configuration in the first place!
Fred Lunnon
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
On 6/29/15, David Wilson <davidwwilson@comcast.net> wrote:
Fred, you really came through. I just knew it had to exist. It's quite beautiful.
Sadly there remain 3 circles adventitiously meeting 4-th points disjoint from the corresponding projective lines! [ I brainstormed earlier over symmetry: 3-fold around one cube diagonal is correct. ] I am currently wrangling its dual (exchanging points with centres), which for some reason is reluctant to display properly; however I strongly suspect that will turn out to present a similar flaw. So although pretty, it's even more thoroughly crocked than FWH's -- for which I intend to post a graphic as soon as I can engineer its point coordinates -- despite it apparently lacking algebraic proof of correctness. WFL
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Fred Lunnon Sent: Monday, June 29, 2015 9:23 AM To: math-fun Subject: Re: [math-fun] Fano Plane puzzle
A couple of (considerably less feeble) views of solid Fano posted at https://www.dropbox.com/s/i4l1szci7o0ww7r/fano_solid0.gif?dl=0 https://www.dropbox.com/s/vpgp26kcr1ttsgk/fano_solid1.gif?dl=0 the first showing 3-fold symmetry about cube diagonal.
WFL
On 6/29/15, Fred Lunnon <fred.lunnon@gmail.com> wrote:
On 6/28/15, David Wilson <davidwwilson@comcast.net> wrote:
Really, all I want is a stabile hanging in my office consisting of circular wire rings of equal radius (in rainbow colors) welded together with Fano connectivity and otherwise not touching one another.
It's not like I'm hard to please. You would think I had asked for circular Borromean rings. Sheesh.
Right then --- here is an elegant embedding exactly as requested (on numerous occasions) by DWW for a Fano plane composed of points and circles. It has 3-fold symmetry about axis line x = y = z , without extraneous collisions or near-misses. Four points lie at alternate corners of a scaled unit cube; the other three at equal displacements along axis lines.
Approximate values for the points and centres are as follows; as before, a unit circle with centre index i (mod 7) meets points indexed i, i+1, i+3 . q := 0.5912088662; p := 1.224744871; points := [ [q,-q,-q], [-q,q,-q], [-q,-q,q], [0,p,0], [0,0,p], [q,q,q], [p,0,0] ]; u := 0.3203726673; v := 0.2819082667; w := 0.4082482903; centres := [ [u,u,-v], [-v,u,u], [-u,u,v], [w,w,w], [v,-u,u], [u,v,-u], [u,-v,u] ];
I have a proof of correctness, and exact algebraic equations for the point components, though not the centres: 2*p^2 - 3 = 0 ; 36*q^4 - 4*q^2 - 3 = 0 ;
Available on request is a Maple worksheet showing a (rather feeble) 3-D manipulable graphic, which ate up an order of magnitude more coding effort than finding the configuration in the first place!
Fred Lunnon
_______________________________________________ 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
Two views of FWH solid circular Fano plane at https://www.dropbox.com/s/iofsjqyln0n929d/hele_solid0.gif?dl=0 https://www.dropbox.com/s/zs7pi91p0f99m5o/hele_solid1.gif?dl=0 the first showing 6-fold symmetry about z-axis, the second with z-axis up page showing one (of 3) adventitous double point below right of centre. No coordinate box this time. For nothing is simple. FWH's circle #1, parallel to xy-plane with centre on z-axis, defeated all attempts to plot it until in desperation I rotated everything thru' an entire radian around all three axes ... Maple solve() moves --- or in this case, fails to go anywhere --- in mysterious ways! WFL On 6/30/15, Fred Lunnon <fred.lunnon@gmail.com> wrote:
On 6/29/15, David Wilson <davidwwilson@comcast.net> wrote:
Fred, you really came through. I just knew it had to exist. It's quite beautiful.
Sadly there remain 3 circles adventitiously meeting 4-th points disjoint from the corresponding projective lines! [ I brainstormed earlier over symmetry: 3-fold around one cube diagonal is correct. ]
I am currently wrangling its dual (exchanging points with centres), which for some reason is reluctant to display properly; however I strongly suspect that will turn out to present a similar flaw.
So although pretty, it's even more thoroughly crocked than FWH's -- for which I intend to post a graphic as soon as I can engineer its point coordinates -- despite it apparently lacking algebraic proof of correctness.
WFL
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Fred Lunnon Sent: Monday, June 29, 2015 9:23 AM To: math-fun Subject: Re: [math-fun] Fano Plane puzzle
A couple of (considerably less feeble) views of solid Fano posted at https://www.dropbox.com/s/i4l1szci7o0ww7r/fano_solid0.gif?dl=0 https://www.dropbox.com/s/vpgp26kcr1ttsgk/fano_solid1.gif?dl=0 the first showing 3-fold symmetry about cube diagonal.
WFL
On 6/29/15, Fred Lunnon <fred.lunnon@gmail.com> wrote:
On 6/28/15, David Wilson <davidwwilson@comcast.net> wrote:
Really, all I want is a stabile hanging in my office consisting of circular wire rings of equal radius (in rainbow colors) welded together with Fano connectivity and otherwise not touching one another.
It's not like I'm hard to please. You would think I had asked for circular Borromean rings. Sheesh.
Right then --- here is an elegant embedding exactly as requested (on numerous occasions) by DWW for a Fano plane composed of points and circles. It has 3-fold symmetry about axis line x = y = z , without extraneous collisions or near-misses. Four points lie at alternate corners of a scaled unit cube; the other three at equal displacements along axis lines.
Approximate values for the points and centres are as follows; as before, a unit circle with centre index i (mod 7) meets points indexed i, i+1, i+3 . q := 0.5912088662; p := 1.224744871; points := [ [q,-q,-q], [-q,q,-q], [-q,-q,q], [0,p,0], [0,0,p], [q,q,q], [p,0,0] ]; u := 0.3203726673; v := 0.2819082667; w := 0.4082482903; centres := [ [u,u,-v], [-v,u,u], [-u,u,v], [w,w,w], [v,-u,u], [u,v,-u], [u,-v,u] ];
I have a proof of correctness, and exact algebraic equations for the point components, though not the centres: 2*p^2 - 3 = 0 ; 36*q^4 - 4*q^2 - 3 = 0 ;
Available on request is a Maple worksheet showing a (rather feeble) 3-D manipulable graphic, which ate up an order of magnitude more coding effort than finding the configuration in the first place!
Fred Lunnon
_______________________________________________ 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 think this is just gorgeous. Is Fred's solution the unique* set of points in R^3 having the combinatorial type of the Fano plane (with unit circles for the lines) and the symmetry group of D_3 == S_3 ??? —Dan _____________________________________ * Up to isometries of R^3, of course.
On Jun 30, 2015, at 11:12 AM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Two views of FWH solid circular Fano plane at https://www.dropbox.com/s/iofsjqyln0n929d/hele_solid0.gif?dl=0 https://www.dropbox.com/s/zs7pi91p0f99m5o/hele_solid1.gif?dl=0 the first showing 6-fold symmetry about z-axis, the second with z-axis up page showing one (of 3) adventitous double point below right of centre.
No coordinate box this time. For nothing is simple. FWH's circle #1, parallel to xy-plane with centre on z-axis, defeated all attempts to plot it until in desperation I rotated everything thru' an entire radian around all three axes ... Maple solve() moves --- or in this case, fails to go anywhere --- in mysterious ways!
On 6/30/15, Dan Asimov <asimov@msri.org> wrote:
I think this is just gorgeous.
Is Fred's solution the unique* set of points in R^3 having the combinatorial type of the Fano plane (with unit circles for the lines) and the symmetry group of D_3 == S_3 ???
—Dan _____________________________________ * Up to isometries of R^3, of course.
An equilateral triangle in 3-space has freedom 7 , a second one with the same axis line freedom 3 , a single point on the axis freedom 1 . Via C_3 symmetry the 7 unit radii impose only 3 constraints; so modulo isometry, a 3-fold symmetric Fano plane has freedom 11 - 3 - 6 = 2 . Since the orbits remain unchanged under D_6 , that would yield the same freedom. Or so I reckon ... WFL
On Jun 30, 2015, at 11:12 AM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Two views of FWH solid circular Fano plane at https://www.dropbox.com/s/iofsjqyln0n929d/hele_solid0.gif?dl=0 https://www.dropbox.com/s/zs7pi91p0f99m5o/hele_solid1.gif?dl=0 the first showing 6-fold symmetry about z-axis, the second with z-axis up page showing one of 3 adventitious double points below right of centre.
No coordinate box this time. For nothing is simple. FWH's circle #4, parallel to xy-plane with centre on z-axis, defeated all attempts to plot it until in desperation I rotated everything thru' an entire radian around all three axes ... Maple solve() moves --- or in this case, fails to go anywhere --- in mysterious ways!
Correction: under D_6 the second equilateral triangle has freedom 2, so a 6-fold symmetric Fano plane has only freedom 1 . WFL On 6/30/15, Fred Lunnon <fred.lunnon@gmail.com> wrote:
On 6/30/15, Dan Asimov <asimov@msri.org> wrote:
I think this is just gorgeous.
Is Fred's solution the unique* set of points in R^3 having the combinatorial type of the Fano plane (with unit circles for the lines) and the symmetry group of D_3 == S_3 ???
—Dan _____________________________________ * Up to isometries of R^3, of course.
An equilateral triangle in 3-space has freedom 7 , a second one with the same axis line freedom 3 , a single point on the axis freedom 1 . Via C_3 symmetry the 7 unit radii impose only 3 constraints; so modulo isometry, a 3-fold symmetric Fano plane has freedom 11 - 3 - 6 = 2 . Since the orbits remain unchanged under D_6 , that would yield the same freedom.
Or so I reckon ...
WFL
That's really nice! How hard would it be to render the rings with seven different colors? On Mon, Jun 29, 2015 at 6:22 AM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
A couple of (considerably less feeble) views of solid Fano posted at https://www.dropbox.com/s/i4l1szci7o0ww7r/fano_solid0.gif?dl=0 https://www.dropbox.com/s/vpgp26kcr1ttsgk/fano_solid1.gif?dl=0 the first showing 3-fold symmetry about cube diagonal.
WFL
On 6/29/15, Fred Lunnon <fred.lunnon@gmail.com> wrote:
On 6/28/15, David Wilson <davidwwilson@comcast.net> wrote:
Really, all I want is a stabile hanging in my office consisting of circular wire rings of equal radius (in rainbow colors) welded together with Fano connectivity and otherwise not touching one another.
It's not like I'm hard to please. You would think I had asked for circular Borromean rings. Sheesh.
Right then --- here is an elegant embedding exactly as requested (on numerous occasions) by DWW for a Fano plane composed of points and circles. It has 3-fold symmetry about axis line x = y = z , without extraneous collisions or near-misses. Four points lie at alternate corners of a scaled unit cube; the other three at equal displacements along axis lines.
Approximate values for the points and centres are as follows; as before, a unit circle with centre index i (mod 7) meets points indexed i, i+1, i+3 . q := 0.5912088662; p := 1.224744871; points := [ [q,-q,-q], [-q,q,-q], [-q,-q,q], [0,p,0], [0,0,p], [q,q,q], [p,0,0] ]; u := 0.3203726673; v := 0.2819082667; w := 0.4082482903; centres := [ [u,u,-v], [-v,u,u], [-u,u,v], [w,w,w], [v,-u,u], [u,v,-u], [u,-v,u] ];
I have a proof of correctness, and exact algebraic equations for the point components, though not the centres: 2*p^2 - 3 = 0 ; 36*q^4 - 4*q^2 - 3 = 0 ;
Available on request is a Maple worksheet showing a (rather feeble) 3-D manipulable graphic, which ate up an order of magnitude more coding effort than finding the configuration in the first place!
Fred Lunnon
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-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
I used my archaic image MGI Photosuite graphics editor to color one of Fred's images. I sent the result to him, perhaps he will put it on the site. If not, I can send to whomever requests by email. Now, who has a 3D printer?
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Mike Stay Sent: Monday, June 29, 2015 7:15 PM To: math-fun Subject: Re: [math-fun] Fano Plane puzzle
That's really nice! How hard would it be to render the rings with seven different colors?
On Mon, Jun 29, 2015 at 6:22 AM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
A couple of (considerably less feeble) views of solid Fano posted at https://www.dropbox.com/s/i4l1szci7o0ww7r/fano_solid0.gif?dl=0 https://www.dropbox.com/s/vpgp26kcr1ttsgk/fano_solid1.gif?dl=0 the first showing 3-fold symmetry about cube diagonal.
WFL
On 6/29/15, Fred Lunnon <fred.lunnon@gmail.com> wrote:
On 6/28/15, David Wilson <davidwwilson@comcast.net> wrote:
Really, all I want is a stabile hanging in my office consisting of circular wire rings of equal radius (in rainbow colors) welded together with Fano connectivity and otherwise not touching one another.
It's not like I'm hard to please. You would think I had asked for circular Borromean rings. Sheesh.
Right then --- here is an elegant embedding exactly as requested (on numerous occasions) by DWW for a Fano plane composed of points and circles. It has 3-fold symmetry about axis line x = y = z , without extraneous collisions or near-misses. Four points lie at alternate corners of a scaled unit cube; the other three at equal displacements along axis lines.
Approximate values for the points and centres are as follows; as before, a unit circle with centre index i (mod 7) meets points indexed i, i+1, i+3 . q := 0.5912088662; p := 1.224744871; points := [ [q,-q,-q], [-q,q,-q], [-q,-q,q], [0,p,0], [0,0,p], [q,q,q], [p,0,0] ]; u := 0.3203726673; v := 0.2819082667; w := 0.4082482903; centres := [ [u,u,-v], [-v,u,u], [-u,u,v], [w,w,w], [v,-u,u], [u,v,-u], [u,-v,u] ];
I have a proof of correctness, and exact algebraic equations for the point components, though not the centres: 2*p^2 - 3 = 0 ; 36*q^4 - 4*q^2 - 3 = 0 ;
Available on request is a Maple worksheet showing a (rather feeble) 3-D manipulable graphic, which ate up an order of magnitude more coding effort than finding the configuration in the first place!
Fred Lunnon
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
It is done: https://www.dropbox.com/s/mngbt1xo6nthr0v/fano_colo0.gif?dl=0 WFL On 6/30/15, David Wilson <davidwwilson@comcast.net> wrote:
I used my archaic image MGI Photosuite graphics editor to color one of Fred's images. I sent the result to him, perhaps he will put it on the site. If not, I can send to whomever requests by email. Now, who has a 3D printer?
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Mike Stay Sent: Monday, June 29, 2015 7:15 PM To: math-fun Subject: Re: [math-fun] Fano Plane puzzle
That's really nice! How hard would it be to render the rings with seven different colors?
On Mon, Jun 29, 2015 at 6:22 AM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
A couple of (considerably less feeble) views of solid Fano posted at https://www.dropbox.com/s/i4l1szci7o0ww7r/fano_solid0.gif?dl=0 https://www.dropbox.com/s/vpgp26kcr1ttsgk/fano_solid1.gif?dl=0 the first showing 3-fold symmetry about cube diagonal.
WFL
On 6/29/15, Fred Lunnon <fred.lunnon@gmail.com> wrote:
On 6/28/15, David Wilson <davidwwilson@comcast.net> wrote:
Really, all I want is a stabile hanging in my office consisting of circular wire rings of equal radius (in rainbow colors) welded together with Fano connectivity and otherwise not touching one another.
It's not like I'm hard to please. You would think I had asked for circular Borromean rings. Sheesh.
Right then --- here is an elegant embedding exactly as requested (on numerous occasions) by DWW for a Fano plane composed of points and circles. It has 3-fold symmetry about axis line x = y = z , without extraneous collisions or near-misses. Four points lie at alternate corners of a scaled unit cube; the other three at equal displacements along axis lines.
Approximate values for the points and centres are as follows; as before, a unit circle with centre index i (mod 7) meets points indexed i, i+1, i+3 . q := 0.5912088662; p := 1.224744871; points := [ [q,-q,-q], [-q,q,-q], [-q,-q,q], [0,p,0], [0,0,p], [q,q,q], [p,0,0] ]; u := 0.3203726673; v := 0.2819082667; w := 0.4082482903; centres := [ [u,u,-v], [-v,u,u], [-u,u,v], [w,w,w], [v,-u,u], [u,v,-u], [u,-v,u] ];
I have a proof of correctness, and exact algebraic equations for the point components, though not the centres: 2*p^2 - 3 = 0 ; 36*q^4 - 4*q^2 - 3 = 0 ;
Available on request is a Maple worksheet showing a (rather feeble) 3-D manipulable graphic, which ate up an order of magnitude more coding effort than finding the configuration in the first place!
Fred Lunnon
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
_______________________________________________ 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
You don't need your own printer for this kind of thing; just upload the .stl to Shapeways. --Michael On Jun 30, 2015 1:56 AM, "Fred Lunnon" <fred.lunnon@gmail.com> wrote:
It is done: https://www.dropbox.com/s/mngbt1xo6nthr0v/fano_colo0.gif?dl=0 WFL
On 6/30/15, David Wilson <davidwwilson@comcast.net> wrote:
I used my archaic image MGI Photosuite graphics editor to color one of Fred's images. I sent the result to him, perhaps he will put it on the site. If not, I can send to whomever requests by email. Now, who has a 3D printer?
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Mike Stay Sent: Monday, June 29, 2015 7:15 PM To: math-fun Subject: Re: [math-fun] Fano Plane puzzle
That's really nice! How hard would it be to render the rings with seven different colors?
On Mon, Jun 29, 2015 at 6:22 AM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
A couple of (considerably less feeble) views of solid Fano posted at https://www.dropbox.com/s/i4l1szci7o0ww7r/fano_solid0.gif?dl=0 https://www.dropbox.com/s/vpgp26kcr1ttsgk/fano_solid1.gif?dl=0 the first showing 3-fold symmetry about cube diagonal.
WFL
On 6/29/15, Fred Lunnon <fred.lunnon@gmail.com> wrote:
On 6/28/15, David Wilson <davidwwilson@comcast.net> wrote:
Really, all I want is a stabile hanging in my office consisting of circular wire rings of equal radius (in rainbow colors) welded together with Fano connectivity and otherwise not touching one another.
It's not like I'm hard to please. You would think I had asked for circular Borromean rings. Sheesh.
Right then --- here is an elegant embedding exactly as requested (on numerous occasions) by DWW for a Fano plane composed of points and circles. It has 3-fold symmetry about axis line x = y = z , without extraneous collisions or near-misses. Four points lie at alternate corners of a scaled unit cube; the other three at equal displacements along axis lines.
Approximate values for the points and centres are as follows; as before, a unit circle with centre index i (mod 7) meets points indexed i, i+1, i+3 . q := 0.5912088662; p := 1.224744871; points := [ [q,-q,-q], [-q,q,-q], [-q,-q,q], [0,p,0], [0,0,p], [q,q,q], [p,0,0] ]; u := 0.3203726673; v := 0.2819082667; w := 0.4082482903; centres := [ [u,u,-v], [-v,u,u], [-u,u,v], [w,w,w], [v,-u,u], [u,v,-u], [u,-v,u] ];
I have a proof of correctness, and exact algebraic equations for the point components, though not the centres: 2*p^2 - 3 = 0 ; 36*q^4 - 4*q^2 - 3 = 0 ;
Available on request is a Maple worksheet showing a (rather feeble) 3-D manipulable graphic, which ate up an order of magnitude more coding effort than finding the configuration in the first place!
Fred Lunnon
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
_______________________________________________ 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
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participants (7)
-
Bill Gosper -
Dan Asimov -
David Wilson -
Ed Pegg Jr -
Fred Lunnon -
Michael Kleber -
Mike Stay