What should our site design look like?

Question

As we are approaching 90 days, it is time to get us back to answering the 7 Essential Questions.

In terms of site design, we ultimately need to consider a number minutiae such as logo, the top banner, the look of our badges, the color scheme used, etc. As this is a large topic, this question should act as a discussion forum for ideas. In particular, we should concentrate on the logo as it will represent us outside of this community.

Answer 1

Inspired by F'x, here's a 2D analogue of spikey. Done with a bunch of rules because I like them, and they make it clear what is going on.

Start with a pentagon:

p = Polygon[Table[N[{Cos[t], Sin[t]}], {t, \[Pi]/10, 2 \[Pi], (2 \[Pi])/5}]];

Make a rule to split any polygon into triangles (centroid to each edge):

triangulate = Polygon[v_] :> (Polygon[Append[#, Mean[v]]] & /@ Partition[v, 2, 1, {1, 1}]);

A rule to split triangles into the spikey face structure:

moretriangles = Polygon[{a_, b_, c_}] :> 
   With[{ab = (a + b)/2, bc = (b + c)/2, ca = (c + a)/2}, {
     Polygon[{a, ab, ca}],
     Polygon[{ab, b, bc}],
     Polygon[{c, ca, ab}],
     Polygon[{c, ab, bc}]
   }];

Put a triangle inside each triangle (which will be done twice):

shrink = Polygon[{a_, b_, c_}] :>
   With[{aa = (6 a + b + c)/8, bb = (a + 6 b + c)/8, cc = (a + b + 6 c)/8}, {
     Polygon[{a, b, bb, aa}],
     Polygon[{b, c, cc, bb}],
     Polygon[{c, a, aa, cc}],
     Polygon[{aa, bb, cc}]
   }];

Add some colour:

colour4 = q : Polygon[{_, _, _, _}] :> {ColorData["NeonColors"][RandomReal[]], q};
colour3 = q : Polygon[{_, _, _}] :> {RGBColor[0.6, 0.12, 0.12], q};

Split each polygon edge into a series of small steps, so we will get curves when we transform it:

curve = Polygon[v_] :> FilledCurve[
    Line[Map[{10 - #, #}/10 &, Range[0, 10]].#] & /@ 
    Partition[v, 2, 1, {1, 1}]];

The hyperbolic transformation:

f[r_] := Re[(ArcSin[2 r - 1] + \[Pi]/2)/2];
bolics = v : {_?NumberQ, _} :> f[Norm[v]] v;

to finally give:

Graphics[p /. triangulate /. moretriangles /. shrink /. shrink /. colour3 /. colour4 /.
         curve /. bolics]

EDIT

Now with better different colours!

Clear[colour4]; 
colour4[s_] := q : Polygon[{_, _, _, _}] :> {ColorData[s][RandomReal[]], q};

SeedRandom[8];
Graphics[p /. triangulate /. moretriangles /. shrink /. shrink /. colour3 /. 
  colour4["SouthwestColors"] /. curve /. bolics]

And some badges with the hyperbolic polygon theme:

Clear[bolics2];
bolics2[s_] := v : {_?NumberQ, _} :> (s f[Norm[v]] + 1 - s) v;
Clear[poly];
poly[start_, n_] := Polygon[Table[N[{Cos[t], Sin[t]}], {t, start, 2 \[Pi], (2 \[Pi])/n}]];
Grid[List[Graphics[{Gray, Translate[#[[2]], {0, -0.2}], #}, ImageSize -> 15] & /@
  {{RGBColor[237/255, 198/255, 47/255], poly[\[Pi]/10, 5] /. curve /. bolics2[1]},
   {RGBColor[218/255, 218/255, 218/255], poly[0, 4] /. curve /. bolics2[2/3]},
   {RGBColor[226/255, 154/255, 84/255], poly[\[Pi]/2, 3] /. curve /. bolics2[1/2]}}]]

---

Addendum by J.M.

I wasn't too fond of our proposed logo having a Random* component, and set about looking for a slightly more systematic coloring scheme. Here is what I came up with:

colour3[s_: LightGray] := q : Polygon[{_, _, _}] :> {s, q}

PolygonCentroid[pts_?MatrixQ] := With[{dif = Map[Det, Partition[pts, 2, 1, {1, 1}]]}, 
  ListConvolve[{{1, 1}}, Transpose[pts], {-1, -1}].dif/(3 Total[dif])]

colour4[s_: "SouthwestColors"] := Polygon[v_] /; Length[v] == 4 :>
        {ColorData[s, (7 Norm[PolygonCentroid[v], 2] - 2)/4], Polygon[v]}


Graphics[p /. triangulate /. moretriangles /. shrink /. shrink /. 
     colour3[] /. colour4[] /. curve /. bolics]

Answer 2

Here's my go at creating a seamlessly repeatable background pattern similar to the dart-kite Penrose tiling. Personally I would prefer the rhombus tiling, but I did not manage to make a periodic approximation to that one yet.

Several people mentioned that it would be nice to have a background based on some sort of Penrose tiling. The problem with Penrose tilings is that they're aperiodic while Jin needs a periodically tileable image for the website's background. So I had to cheat a little to make the pattern repeatable.

First, here's a sample of the result (please click to magnify!):

Edit: here's an alternative made using draw2:

This is the "base cell" that can be repeated in a RЯRЯRЯRЯRЯRЯ fashion horizontally only to create a fake Penrose tiling that looks convincing enough. If you look closely enough you'll notice that some of the tiles are neither a kite nor a dart but a parallelogram---an artefact of making the pattern periodic.

Another possibly interesting variation which I did not have time to implement as vector art (it's easier using image processing) is to trace out the boundary between the light and dark regions only (and not the boundary of each kite and dart).

---

The code follows at the end. It's based on 'deflation'. Please "steal" the code, make other variations, and post them! If you need me to explain some part of the code, post a comment, please!

(* rotate vector by 90 deg *)
rot90[{x_, y_}] := {-y, x}

(* divide the segment AB using ratio R *)
div[a_, b_, r_] := a + (b - a) r

deflate =
  {ki[a_, b_, o_: 1] :>
    With[
     {c = (a + b)/2 + o rot90[b - a] 1/2 Tan[72 Degree]},
     With[
      {d = div[c, a, 1/GoldenRatio],
       e = div[b, c, 1/GoldenRatio]},
      {ki[d, a, o], ki[d, e, -o], da[c, d, o]}
      ]
     ],
   da[a_, b_, o_: 1] :>
    With[
     {c = (a + b)/2 + o rot90[b - a] 1/2 Tan[36 Degree],
      d = div[a, b, 1/GoldenRatio]},
     {ki[c, d, -o], da[b, c, o]}
     ]
   };

(* kite and dart colours *)
kico = GrayLevel[0.93];
daco = GrayLevel[0.85];

draw =
  {ki[a_, b_, o_: 1] :>
    With[
     {c = (a + b)/2 + o rot90[b - a] 1/2 Tan[72 Degree]},
     {kico, Polygon[{a, b, c}]}
     ],
   da[a_, b_, o_: 1] :>
    With[
     {c = (a + b)/2 + o rot90[b - a] 1/2 Tan[36 Degree]},
     {daco, Polygon[{a, b, c}]}
     ]
   };

(* line colour *)
lico = GrayLevel[0.65]

(* draw the outlines of tiles too *)
draw2 =
  {ki[a_, b_, o_: 1] :>
    With[
     {c = (a + b)/2 + o rot90[b - a] 1/2 Tan[72 Degree]},
     {kico, Polygon[{a, b, c}], {lico, Line[{a, b, c}]}}
     ],
   da[a_, b_, o_: 1] :>
    With[
     {c = (a + b)/2 + o rot90[b - a] 1/2 Tan[36 Degree]},
     {daco, Polygon[{a, b, c}], {lico, Line[{a, b, c}]}}
     ]
   };


(* apply the deflation n times *)
defl[expr_, n_] := Nest[# /. deflate &, expr, n] /. draw

(* for "straightening" the pattern *)
r = RotationTransform[-18. Degree];

g = Graphics[
  {EdgeForm[None], 
   defl[N@{ki[r@{0, 0}, r@{1, 0}], 
      ki[r@{1/2, 1/2 Tan[72 Degree]}, r@{-1/2, 1/2 Tan[72 Degree]}]}, 
    9]}
  ]

(* rasterize the graphics, avoiding visible seams between polygons *)
rast = ImageResize[
    Rasterize[Style[#, Antialiasing -> False], "Image", 
     ImageResolution -> 3 72], Scaled[1/3]] &;

img = rast[Show[g, ImageSize -> 400]]

(* crop and assemble the tile *)
img = ImageCrop[img];

ImageCrop[
 ImageAssemble[
  Table[Unevaluated@
    Sequence[img, ImageReflect[img, Left -> Right]], {3}]], {Full, 
  200}]

Answer 3

I'm usually of the opinion that the job of design should be left to professional designers, because they (ought to) know what they're doing. Of course, we can all look at designs around us and say to ourselves "I could do better than that!", but our tastes are so different: I love the things that you hate, and vice versa. (Consider the logo for the 2012 London Olympics - http://blog.gale.com/speakingglobally/the-view-from-here/london-2012-olympics-logo-an-emblem-of-controversy/ - a lot of people really dislike it.)

Having said all that, I'm happy to join in the fun and post ideas for logos and graphics for the Mathematica site. If nothing else, some practice in criticising and abusing designs should come in handy when the real designers move in... :)

Rationale: It's supposed to be simple (and should work as an icon), it's made of gold (although it looks a bit more coppery than I expected), and it's definitely not a spikey, although it is made with Mathematica and shows some of Mathematica's image and 3D graphics abilities.

Answer 4

I think it would be a nice and useful addition, fitting the site's infrastructure, to have some of the built-in styles of Mathematica here. I'm not sure this would be approved by SE, as it is possible, that they only allow for one type of quote-style and one type of code-style, but certainly it is possible to implement. For example, quotation from the online documentation should look like this:

Of course, uploading an image always works, but is a waste of bandwidth. I also find the dotted horizontal divider useful to visually separate examples:

Answer 5

Since the original logo has turned out to be a bit too close to the Mathematica logo for Wolfram's lawyers' comfort, here are a few alternatives that maintain the essential look and color palette, but are different enough to hopefully skirt around the problem.

Of course, we can have any combination of number of points, stellation, and additional complexity, and the colors can be tweaked to suit.

I am using the same code as in J.M.'s post, except that:

colour4[s_: "SunsetColors", a_?NumericQ, b_?NumericQ] := Polygon[v_] /; 
  Length[v] == 4 :> {ColorData[s, a - b Norm[PolygonCentroid[v]]], 
  Polygon[v]}

tweakable curvature:

bolicsn[n_] := 
  v : {_?NumberQ, _} :> v Re[(ArcSin[2 Norm[v] - 1] + Pi/n)/2];

and the starter code can be any one of:

p4 = Polygon[Table[N[{Cos[t], Sin[t]}], {t, Pi/10, 2 Pi, 2 Pi/4}]];
p5 = Polygon[Table[N[{Cos[t], Sin[t]}], {t, Pi/10, 2 Pi, 2 Pi/5}]];
p6 = Polygon[Table[N[{Cos[t], Sin[t]}], {t, Pi/6, 2 Pi, 2 Pi/6}]];
p7 = Polygon[Table[N[{Cos[t], Sin[t]}], {t, Pi/14, 2 Pi, 2 Pi/7}]];
p7b = Polygon[Table[N[{Cos[t], Sin[t]}], {t, 3 Pi/14, 2 Pi, 2 Pi/7}]];
p8 = Polygon[Table[N[{Cos[t], Sin[t]}], {t, Pi/4, 2 Pi, 2 Pi/8}]];

Seven-point alternative

I think a seven-point alternative will be less likely to attract problems, especially if it is more purple and gold than orange-red, and less spiky. Some alternatives:

Graphics[p7b /. triangulate /. moretriangles /. shrink /. shrink /. 
 colour3[] /. colour4["SunsetColors", 9/8, 31/34] /. curve /. 
 bolicsn[0.4], ImageSize -> 400]

Or

Graphics[p7b /. triangulate /. moretriangles /. shrink /. shrink /. 
 shrink /. colour3[] /. colour4["SunsetColors", .95, 32/34] /. 
 bolicsn[0.4], ImageSize -> 400]

Older versions

Example 1: Pentagon, not hyperbolic

Graphics[p5 /. triangulate /. moretriangles /. shrink /. shrink /. 
shrink /. colour3[] /. colour4[], ImageSize -> 400]

Example 2: Heptagon, not hyperbolic

Graphics[p7 /. triangulate /. moretriangles /. shrink /. shrink /. 
shrink /. colour3[] /. colour4["SunsetColors", 1, 28/34], ImageSize -> 400]

Example 3: Octagon, hyperbolic

Graphics[p8 /. triangulate /. moretriangles /. shrink /. shrink /. 
 colour3[] /. colour4["SunsetColors", 9/8, 31/34] /. curve /. bolics, ImageSize -> 400]

Example 4: A fussier non-hyperbolic pentagon
This probably needs the mesh lines to be turned off, which I haven't worked out how to do.

Graphics[p5 /. triangulate /. triangulate /. moretriangles /. 
   shrink /. shrink /. shrink /. colour3[] /. Mesh -> None /. 
   colour4["SunsetColors", 1, 30/34], ImageSize -> 400]

Example 5: less curved version of the original pentagon

Graphics[{p5 /. triangulate /. moretriangles /. shrink /. shrink /. 
  colour3[] /. colour4[] /. curve /. bolicsn[0.75]}, ImageSize -> 400]

Answer 6

Another approach to get something like a Penrose pattern which can be used for a tiled image is to start from a square.

These triangles don't follow the golden ratio rule which is usually required for Penrose patterns, but the iteration can be used anyway. Using these triangles as half-darts, one could now start to generate the subtriangles following the rules of deflation for P2 and P3 tilings.

The good thing here is, that this tile is instantly repeatable in any direction.

Implementation

The code is really simple. Just look in the above link that there are two kinds of triangles. Every triangle has its own subdivision rule which is just a combination of the sides and one or two new points on some triangle-sides. Using t1 for the half-kites and t2 for the half-dartes the subdivision rule is

subdivide = {
   t1[{a_, b_, c_}] :> 
    With[{p1 = a + (b - a)/GoldenRatio, 
      p2 = c + (a - c)/GoldenRatio},
     {t2[{p2, p1, a}], t1[{c, p1, b}], t1[{c, p1, p2}]}], 
   t2[{a_, b_, c_}] :> With[{p = c + (b - c)/GoldenRatio},
     {t1[{c, a, p}], t2[{p, a, b}]}
     ]
   };

The used points a,b,c are here complex numbers and if you want to draw them, you take real and imaginary part.

draw = {
   t1[arg_] :> {GrayLevel[.95], Polygon[{Re[#], Im[#]} & /@ arg]}, 
   t2[arg_] :> {GrayLevel[.85], Polygon[{Re[#], Im[#]} & /@ arg]}
   };

That's it. More is in the basic version not required. The only thing which is missing is some initial triangle or set of triangles. The rest is easy. Take the initial triangle(s) and apply the sub-division rule as often as you want, use the draw rule at the end and wrap Graphics around it.

With[{initial = t1[{0.0, Exp[-I*Pi/10.0], Exp[I*Pi/10.0]}]},
 Graphics[Nest[# /. subdivide &, initial, 6] /. draw]
 ]

Remark 1: the first coordinate in t1 is always the top corner (36 deg) and in t2 right corner (108 deg) of the triangles in this image. The order of the last two points of the triangles does matter, because the next subdivision step is not symmetric. For instace, in my first square image above, the center point is always the first coordinate in each triangle but the other two points are specified in a way that neighboring triangles have not the same circumferential direction.

Remark 2: Since until the draw rule is applied we have complex numbers, it is really easy to use most transformations like translation, rotation, etc. A rule for translation and rotation is just

trans[p_] := (h : (t1 | t2))[pts_] :> h[pts + p];
rot[phi_] := (h : (t1 | t2))[pts_] :> h[pts*Exp[I phi]];

With the first triangulated square (pts) you can create a tiling by using the trans rule in a table

With[{pts = Table[
    t2[{0, Exp[I (phi - Pi/4.0)], 
       Exp[I (phi + Pi/4.0)]}] /. (h : (t1 | t2))[{a_, b_, 
        c_}] :> {h[{b + (c - b)/2, a, b}], h[{c + (b - c)/2, a, c}]},
    {phi, 0, 2 Pi, Pi/2}]},
 Graphics[{EdgeForm[{GrayLevel[.76]}], 
   Nest[# /. subdivide &, 
     Table[pts /. trans[i*Sqrt[2]] /. trans[j*I*Sqrt[2]], {i, 10}, {j,
        5}], 2] /. draw}, ImageMargins -> None, 
  PlotRangePadding -> None]
 ]

Or you can create a Penrosed version of our logo by using a subdivided pentagon and the nonlinear transformation (bolics above)

Answer 7

I have seen all these beautiful ideas dedicated to the logo and decided to add another one. And though it maybe already too late I still would like to post it – just to share and add to the pool of creative approaches. My train of thought was the following. Indeed Mathematica graphics is stunning and the logo should reflect on that. Yet the nature of the Mathematica functional language itself is very unique. Incidentally I also liked the simplicity of Apple SE idea - Apple command key is a unique keyboard symbol that indeed reminds many people instantly – “This is Apple stuff, yes, for sure…” But we have all these @,#,&,~, etc. that are famous entities of our programming meta-culture. Also Mathematica for me and many others is a creative discovery tool, something that allows ideas to emerge and solidify handsomely. So to summarize I was looking for something reflecting upon:

• Unique simple logo reflecting on unique functional language
• Creative discovery tool, idea emergence
• Stunning graphics

I acknowledge it is probably not that “unique, creative, stunning”, yet here it is – some attempt:

This is the code to play with different designs. The code needs to be adjusted to scale well for smaller sizes. This can be done I think, please ask if you need help.

Manipulate[
 SeedRandom[rd];

 ListDensityPlot[(MapThread[
     Append, {RandomReal[{-.1, 1.1}, {800, 2}], 
      RandomReal[{0, .5 - s}, 800]}]~Join~
    MapThread[
     Append, {Rescale@pdata[[1 ;; -1 ;; 6]], 
      RandomReal[{.5 + s, 1}, Length[pdata[[1 ;; -1 ;; 6]]]]}]), 
  InterpolationOrder -> 0, Mesh -> All, Frame -> False, 
  ColorFunction -> gradients, ImageSize -> 600, PlotRange -> All, 
  PlotRangePadding -> 0]

 , "randomize"
 , {{rd, 7, ""}, 1, 1000, 1, ImageSize -> Small}
 , "thickness"
 , {{s, .1, ""}, -.5, .5, ImageSize -> Small}
 , {{gradients, "DeepSeaColors", 
   ""}, (# -> Show[ColorData[#, "Image"], ImageSize -> 100]) & /@ 
   ColorData["Gradients"]}
 , FrameMargins -> 0, ControlPlacement -> Left, 
 SynchronousUpdating -> False,
 Initialization :> (pdata = 
     Position[ImageData[Binarize@ImageRotate[ColorNegate[
          Rasterize[Style["/@", FontFamily -> "Times"], 
            ImageSize -> 200] // Image], -Pi/2]], 1];)]

Also it could be something as simple as beautified text. Here is our Identity function – just to symbols ;) Well that's just some raw ideas. I am sure it is too late anyway. The code that makes the text is at the end.

text1 = First[First[ImportString[
     ExportString[Style["#", FontSize -> 24, FontFamily -> "Times"], 
      "PDF"], "PDF", "TextMode" -> "Outlines"]]];

text2 = First[First[ImportString[
     ExportString[Style["&", FontSize -> 24, FontFamily -> "Times"], 
      "PDF"], "PDF", "TextMode" -> "Outlines"]]];

GraphicsRow[{Graphics[{EdgeForm[Opacity[.2]], 
    Table[{ColorData["TemperatureMap"][t], 
      Rotate[Translate[text1, 4 {-t, t}], -t/2]}, {t, 0, 1, 1/10}]}, 
   ImageSize -> Medium], 
  Graphics[{EdgeForm[Opacity[.2]], Table[{ColorData["TemperatureMap"][t], 
      Rotate[Translate[text2, 4 {-t, t}], t/2]}, {t, 0, 1, 1/10}]}, 
   ImageSize -> Medium]}, Spacings -> 0]

Answer 8

Far too late to the party but here are my 2c worth. I like the idea of using bracket or other typography unique to MMa. However, I was playing with the idea of using a mathematical construct like a Minimal Surface and rotate it so it 2D appearance results in some interesting geometry. Examples below.

Or more symetrical. The meshing might not work well at all sizes and the colour scheme is just standard MMa but that might make it more recognizable. At least I always recognize a 3D graph with MMa if I see it in the standard colouring scheme.

Edit I should also provide the code for this:

Show[ParametricPlot3D[{u - (u^3/3) + u v^2, v - (v^3/3) + u^2 v, 
   u^2 - v^2}, {u, -2, 2}, {v, -2, 2}, Boxed -> False, Axes -> False, 
  MeshStyle -> Thickness[0.003], 
  BoundaryStyle -> Directive[Black, Thickness[0.008]], 
  PlotPoints -> 50], ImageSize -> 800]

And here another perspective. Quite versatile this object.

As pointed out in the comments below, thanks, this is one of Ennepers Minimal Surfaces. Details can be found here: Weisstein, Eric W. "Enneper's Minimal Surface." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/EnnepersMinimalSurface.html

Answer 9

Attempt 1

Instead of the hexgrid in the initial design, I thought we could consider a tiling pattern that fits in with the spiky theme.

curve = Polygon[v_] :>    FilledCurve[
    Line[Map[{10 - #, #}/10 &, Range[0, 10]].#] & /@ 
     Partition[v, 2, 1, {1, 1}]];

bolics = v : {_?NumberQ, _} :> v Re[(ArcSin[2 Norm[v] - 1] + Pi/2)/2];

p = Polygon[Table[N[{Cos[t], Sin[t]}], {t, Pi/6, 2 Pi, 2 Pi/6}]];
pp = Polygon[Table[N[{Cos[t], Sin[t]}], {t, 0, 2 Pi, 2 Pi/6}]];

mini = Graphics[ Join[{AbsoluteThickness[2], 
     GrayLevel[0.91]}, (First@(p /. curve)) /. bolics], ImageSize -> 50];

minib = Graphics[ Join[{AbsoluteThickness[2], 
     GrayLevel[0.91]}, (First@(pp /. curve)) /. bolics], ImageSize -> 50];

row1 = PadLeft[{mini, minib}, 10, {mini, minib}];
row2 = PadLeft[{minib, mini}, 10, {minib, mini}];

tile = GraphicsGrid[PadLeft[{row1, row2}, 10, {row1, row2}], 
  ImageSize -> 300, Spacings -> {Scaled[-0.1], Scaled[-0.2]}, Background -> GrayLevel[0.95]]

I'm not 100% satisfied this - it looks weirdly irregular, even though it isn't.

Attempt 2

This demonstration can create nice tilings that can probably be cropped to be proper tiles.

I downloaded the code, changed the line defining colors to read:

{cK, cD, lK, lD} = {GrayLevel[0.95], GrayLevel[0.93], GrayLevel[0.97],
  GrayLevel[0.97]}

And then took a snapshot which I modified as below.

DynamicModule[{dec = 1, n = 6}, 
 Graphics[{EdgeForm[{Thin, 
     GrayLevel[
      0.8]}], (kite[0, #1, n, 
       dec] &) /@ {1, -\[Zeta], \[Zeta]^2, -\[Zeta]^3, -1 + \[Zeta] - \
\[Zeta]^2 + \[Zeta]^3}}, ImageSize -> {500, 400}]]

Answer 10

These are just preliminary thoughts after a long flight, which I'll update as time permits.

• As well as avoiding the "spikey" look to avoid trademark issues, we should avoid replicating the canonical color schemes Wolfram uses. Nowadays that is black with red-oranges as seen in the website; red-orange is also the spikey's color. We could use purples, along the lines of the old Mathematica Book covers, or something completely different. Dark greens and coppery colors would go well with cormullion's proposed logo.

• elements such as badge shapes could also be based on (2D representations of) variants on spikeys.

• Cell bracket type styling on the right hand side of posts would be a nice touch.

Answer 11

This would be a first attempt at a code that generates the knot. Graphics experts can prettify it

With[{absRoundabouts = 5, phaseRoundabouts = 2, depthRoundabouts = 5},
  ParametricPlot3D[
   Evaluate@
    Append[(2 + Cos[2 Pi t absRoundabouts]) Through@{Cos, Sin}[
        2 Pi phaseRoundabouts t], Sin[2 Pi t depthRoundabouts]], {t, 
    0, 1}, Boxed -> False, Axes -> False, 
   ViewVector -> {{0, 0, 15}, {0, 0, 0}}] /. 
  Line[pts_, rest___] :> Tube[0.8 pts, 0.2, rest]
 ]

It's easy to play around to make it more spikey

With[{absRoundabouts = 7, phaseRoundabouts = 3, depthRoundabouts = 7},
  ParametricPlot3D[
   Evaluate@
    Append[(2 + Cos[2 Pi t absRoundabouts]) Through@{Cos, Sin}[
        2 Pi phaseRoundabouts t - 0.5 Sin[2 Pi absRoundabouts t]], 
     Sin[2 Pi t depthRoundabouts]], {t, 0, 1}, Boxed -> False, 
   Axes -> False, ViewVector -> {{0, 0, 15}, {0, 0, 0}}] /. 
  Line[pts_, rest___] :> Tube[0.8 pts, 0.2, rest]
 ]

With[{absRoundabouts = 11, phaseRoundabouts = 3, 
  depthRoundabouts = 11}, 
 ParametricPlot3D[
   Evaluate@
    Append[(2 + Cos[2 Pi t absRoundabouts]) Through@{Cos, Sin}[
        2 Pi phaseRoundabouts t - 0.3 Sin[2 Pi  absRoundabouts t]], 
     Sin[2 Pi t depthRoundabouts]], {t, 0, 1}, Boxed -> False, 
   Axes -> False, ViewVector -> {{0, 0, 15}, {0, 0, 0}}] /. 
  Line[pts_, rest___] :> Tube[0.8 pts, 0.1, rest]
 ]

Answer 12

Since Jin has already put up the (nearly) final design for the main site, here's the final code for producing our official logo, as adapted from the code by wxffles and the coloring by Verbeia (which is what I used when I sent Jin the *.eps logo file):

p = Polygon[Table[N[{Cos[t], Sin[t]}], {t, Pi/10, 2 Pi, 2 Pi/5}]];

triangulate = Polygon[v_] :> (Polygon[Append[#, Mean[v]]] & /@ Partition[v, 2, 1, {1, 1}]);

moretriangles = Polygon[{a_, b_, c_}] :>
      With[{ab = (a + b)/2, bc = (b + c)/2, ca = (c + a)/2},
           {Polygon[{a, ab, ca}], Polygon[{ab, b, bc}],
            Polygon[{c, ca, ab}], Polygon[{c, ab, bc}]}];

shrink = Polygon[{a_, b_, c_}] :> 
      With[{aa = (6 a + b + c)/8, bb = (a + 6 b + c)/8, cc = (a + b + 6 c)/8},
           {Polygon[{a, b, bb, aa}], Polygon[{b, c, cc, bb}],
            Polygon[{c, a, aa, cc}], Polygon[{aa, bb, cc}]}];

colour3[s_: LightGray] := q : Polygon[{_, _, _}] :> {s, q};

PolygonCentroid[pts_?MatrixQ] := With[{dif = Map[Det, Partition[pts, 2, 1, {1, 1}]]}, 
  ListConvolve[{{1, 1}}, Transpose[pts], {-1, -1}].dif/(3 Total[dif])]

colour4[s_: "SunsetColors"] := Polygon[v_] /; Length[v] == 4 :>
        {ColorData[s, 8/7 - 35/34 Norm[PolygonCentroid[v]]], Polygon[v]}

curve = Polygon[v_] :> 
  FilledCurve[Line[Map[{10 - #, #}/10 &, Range[0, 10]].#] & /@ 
    Partition[v, 2, 1, {1, 1}]];

bolics = v : {_?NumberQ, _} :> v Re[(ArcSin[2 Norm[v] - 1] + Pi/2)/2];

Graphics[p /. triangulate /. moretriangles /. shrink /. shrink /.
         colour3[] /. colour4[] /. curve /. bolics, ImageSize -> Full]

Export["mmaSELogo.eps", %]

Answer 13

I'm sitting in an airport waiting for a flight that might never come… so I play with Mathematica.

If we want to go with something that is both “spiky” and “colorful”, some 2D ideas could be fun. I have a naïve example to express what I mean:

Show[
 DensityPlot[
  Sin[6*ArcTan[x, y]]*(x^2 + y^2)^0.2, {x, -1.2, 1.2}, {y, -1.2, 
   1.2},
  ColorFunction -> Hue, PlotPoints -> 201,
  RegionFunction -> 
   Function[{x, y}, 
    Sqrt[x^2 + y^2] < 0.2 + Exp[-Abs[Sin[4*ArcTan[x, y]]]^0.7]]],
 PolarPlot[0.2 + Exp[-Abs[Sin[4*t]]^0.7], {t, 0, 2 \[Pi]}, 
  PlotStyle -> Directive[Gray, Thickness[0.015]]],
 PolarPlot[0.2 + Exp[-Abs[Sin[4*t]]^0.7], {t, 0, 2 \[Pi]}, 
  PlotStyle -> Directive[Black, Thick]],
 Frame -> False
 ]

Answer 14

I think good design elements (beyond just the logo) would be things that are recognizable features of Mathematica. For example:

• cell brackets
• In/Out labels
• the default lighting for 3D surfaces
• typesetting
• idiomatic fonts and characters (\[ExponentialE], \[ImaginaryI], \[DifferentialD], etc...)
• code in Courier
• default styles for Plot (ColorData[1])
• auto-italicise words, like Mathematica

Answer 15

One think I like about tex.SE's design is the centered title (as opposed to the left-corner title on most SE sites, including Mathematica beta) with the tex-typical braces around them. I think we could do the same for Mathematica, using the double-struck brackets, i.e. "〚Mathematica〛" large centered at the top (like "{TeX}" for tex.SE).

In case it doesn't display correctly for you: 〚 is \[LeftDoubleBracket] and 〛 is \[RightDoubleBracket].

Another thought: We could use the authentic Mathematica fonts in the preformatted code sections where installed (as far as I understand it, SE sites don't provide fonts for display — which may be problematic with Mathematica font licensing — but rely on installed fonts; since HTML allows to specify fallback fonts, we could therefore display Mathematica fonts where installed, and standard fonts otherwise, without any licensing problems).

Answer 16

Icon design

For fun I took the existing Stack Exchange site icons:

(Some of them you could never guess if you didn't know.)

There's a few that could be seen as Starry Symmetrical Thing, so perhaps something more asymmetrical would be good for Mathematica's icon. How about a big M?

Also, having a distinctive color makes recognition easier.