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I'm a graphics guy and a wanna-be mathematician. Is the T-shirt design below okay? Or if there's a bone headed error, I'd appreciate a heads up.

Pi squared over twelve

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closed as off-topic by Morgan Rodgers, Sahiba Arora, Claude Leibovici, user99914, mathreadler Jan 6 '18 at 19:27

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question is not about mathematics, within the scope defined in the help center." – Morgan Rodgers, Sahiba Arora, Claude Leibovici, Community, mathreadler
If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ @Jahambo99 The sum is the area that is black. $\endgroup$ – Milo Brandt Dec 21 '16 at 23:10
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    $\begingroup$ Somehow the above numbers and the picture fail to illustrate (for me) why the sum is true. $\endgroup$ – Andres Mejia Dec 21 '16 at 23:12
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    $\begingroup$ The $-1$ should definitely be in parentheses. $-1^{n+1}=-1$ for all $n$, while $(-1)^{n+1}$ alternates between $+1$ and $-1$. $\endgroup$ – celtschk Dec 21 '16 at 23:13
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    $\begingroup$ @Mejia I don't think the picture is supposed to constitute any sort of proof $\endgroup$ – Omnomnomnom Dec 21 '16 at 23:13
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    $\begingroup$ Also, slightly off-topic because it's more typography than mathematics: Using a hyphen as minus sign doesn't look nice. $\endgroup$ – celtschk Dec 21 '16 at 23:22
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I would try and optimize the sum at the bottom a bit. Definitely we need parentheses around the $-1$, but just throwing them in will cause a bit of a mess:

$$\pi^2/12 = \sum_{n=1}^\infty ((-1)^{n+1}(1/n^2))$$

Plus, math that's not typeset well already gives me a less-than-stellar vibe; I'm not a huge fan of the font there.

Instead, you could rewrite the sum a bit, if you have the space for it. The version below takes up only slightly more vertical space than the Sigma, if vertical space is a concern. It's a bit less cluttered, and LaTeX's job (technically Mathjax here) is to make sure the typesetting is as good as possible.

$$\displaystyle \frac{\pi^2}{12} = \sum_{n=1}^\infty \dfrac{(-1)^{n+1}}{n^2}$$

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    $\begingroup$ Note that you can click the "Edit" button at the bottom of the answer here to see how things were typeset, the syntax. $\endgroup$ – pjs36 Dec 21 '16 at 23:20
  • $\begingroup$ Thank you pjs36! I'm going to use the first version since I'm constrained by the screens my shirt printer uses. It can't be more than 16" tall. Therefore I will use the version that takes less vertical space. $\endgroup$ – HopDavid Dec 21 '16 at 23:24
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    $\begingroup$ Sure! And just to be clear, I do like things overall -- especially how even the curly braces at the top all line up well; the transition from end-of-brace to dashed-line are all on a line. But fawning doesn't make a very good answer! :P $\endgroup$ – pjs36 Dec 21 '16 at 23:27
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    $\begingroup$ You can also get the $\rm\LaTeX$ form (or a MathML version, in case that for your design you use a program that handles that) by right-clicking on the MathJax formula and selecting the appropriate entry of the "Show Math As" submenu. $\endgroup$ – celtschk Dec 21 '16 at 23:28
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    $\begingroup$ You can save a lot of vertical space by using vertical bars and horizontal arrows around your numbers (somewhat like this drawing: me.umn.edu/courses/me2011/handouts/drawing/index_files/…) instead of those huge curly braces. It could also be more legible. That should more than make up for the space required to typeset the sum nicely. (Notice how it's actually the summation symbol that takes up the most vertical space, even in the last formula; writing the fractions vertically doesn't make any difference in the height.) $\endgroup$ – David K Dec 22 '16 at 8:44
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Yet another pleasing way to write the sum:

$$ \left. \pi^2\big /12\right. = \sum_{n=1}^\infty (-1)^{n+1}\big/n^2 $$

say, if you wanted to take more horizontal space.

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  • $\begingroup$ I like that! But I've already had the silk screener burn the stencil. $\endgroup$ – HopDavid Dec 22 '16 at 17:23
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    $\begingroup$ @HopDavid So why did you ask for feedback if you already have the stencil made? $\endgroup$ – Mast Dec 23 '16 at 12:14
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    $\begingroup$ @Mast Ummm…. No. I asked for feedback. pjs36 and others gave it, and I put the -1 in parenthesis. Then I gave the corrected design to my silkscreener and asked him if he could get me at least 1 shirt before Christmas. Then I saw the above answer. $\endgroup$ – HopDavid Dec 23 '16 at 17:30
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I prefer to centralise elements that sit on ground level, i.e. the "footprint", and ignore cantilevered superscripts, e.g. $$\huge\frac {\;\; \pi^{^2}}{12}\qquad,\qquad \frac 1{\; n^{^2}}$$ which seem more pleasing aesthetically than the standard $$\huge\frac {\pi^2}{12} \qquad,\qquad \frac 1{n^2}$$

If we do that then we have $$\huge\color{darkblue}{\boxed{\frac {\;\;\;\; \pi^{^2}}{12}=\sum_{n=1}^\infty (-1)^{^{n+1}}\frac1{\;\;n^{^2}}}}$$ or, using default font size for superscripts,
$$\huge\color{darkblue}{\boxed{\frac {\;\;\;\; \pi^2}{12}=\sum_{n=1}^\infty (-1)^{n+1}\frac1{\;\;n^2}}}$$

I prefer to separate the $(-1)^{n+1}$ so that it doesn't make the fraction top heavy.

I also prefer that fractions be shown as $\frac ab$ rather than $a/b$ as they are being used in display mode rather than an in-line mathematical expression or a subscript/superscript.

And, of course, for typesetting mathematics, a font with serifs always looks better. Use the image from mathjax here, from the equation writer in Microsoft Word, or from other mathematical typesetting software like $\TeX$.


Note

It might be interesting to consider the comparison in the picture below, taken from chalkdustmagazine.com here.

$\hspace{4cm}$ enter image description here

According to the website, the six examples are:

  1. Official Chalkdust font editor’s handwriting, using black Parker fountain pen
  2. Professional typesetting, from a 1954 UCL maths exam paper,
  3. Typewriter, from a 1985 exam paper, in a monospaced font,
  4. Microsoft’s pre-2007 equation editor, using Times New Roman,
  5. Microsoft’s post-2007 equation editor, using Cambria Math,
  6. LaTeX, using Computer Modern.

Notice that in example 1 (handwritten), the $1$ and $n$ are centred, and the superscript $2$ is much smaller than $n$. Hence when writing, it seems to be natural to do this.

For example 2 (manual typesetting), ite seems that the typesetter is trying to centre the $1$ and $n$ as much as possible, although the $2$ is almost the same size as $n$.

For examples 4,5,6 (all computer typesetting systems), the $n$ is off-centre. However, for example 4, the spacing and font size of superscript $2$ seems to be much more pleasing in that it is placed higher than $n$ and is also smaller in size.

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    $\begingroup$ Your superscript 2 in $\pi^2$ is bigger than the superscript 2 in $n^2$, which looks odd to me. $\endgroup$ – Nathaniel Dec 23 '16 at 7:21
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    $\begingroup$ Yes you noticed that. Somehow I find that the standard font size for $2$ in $\frac 1{n^2}$ is too large and almost the same size as $n$. However, when the same is applied to $\frac {\pi^2}{\cdot}$ it looks ok. $\endgroup$ – hypergeometric Dec 23 '16 at 7:31
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    $\begingroup$ For each fraction individually I agree with you, but I think being inconsistent looks worse than using the standard $n^2$. You could use small superscripts for all of them I suppose. $\endgroup$ – Nathaniel Dec 23 '16 at 7:47
  • $\begingroup$ @Nathaniel - Good. Let's see how that looks. $\endgroup$ – hypergeometric Dec 23 '16 at 7:57
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    $\begingroup$ It's so hard not to keep nit-picking... now it looks strange because (i) there is a vertical gap between the top of the $pi$ and the baseline of the $2$, which isn't there in the $n^2$ expression, and (ii) because the superscript in $(-1)^{n+1}$ is still in the original size, as are the expressions above and below the summation sign. For point (i) I'm not really sure why that's happening, since the height of the $\pi$ character looks essentially the same as that of the $n$. I think now I'd hedge towards using the original superscript size throughout, while keeping the centred fractions. $\endgroup$ – Nathaniel Dec 23 '16 at 8:04

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