two different \mathcal{H}

Question

I encountered the following version of \mathcal{H} and like it a lot:

This is slightly different from the usual one from Computer Modern font:

I have done some search and tried different fonts, e.g., mtpro2, mathptmx, and mathpazo etc., but could not find the first \mathcal{H}.

Here is my question: Is there a way that I could get the first one? Is it some sort of "old \mathcal symbols"?

Any help would be highly appreciated.

PS: I have found an example for the source of the first \mathcal{H} here is the link on arXiv preprint page.

I downloaded the *.tex file of that pdf, and here are all the packages used there:

\documentclass[11pt]{article}  
\usepackage{xcolor}  
\definecolor{labelkey}{rgb}{0,0.08,0.45}  
\definecolor{refkey}{rgb}{0,0.6,0.0}  
\definecolor{Brown}{rgb}{0.45,0.0,0.05}  
\definecolor{dgreen}{rgb}{0.00,0.40,0.00}  
\definecolor{dblue}{rgb}{0,0.08,0.45}  

\usepackage{amsmath}  
\usepackage{amssymb}  
\usepackage{theorem}

Of course, typing \mathcal{H} in this setting yields the second one, but somehow the \mathcal{H} in the link looks like this:

So I guess there must be something "weird" going on here.

PS: I often see the first one in papers published by Springer, e.g., https://link.springer.com/content/pdf/10.1007%2Fs000200300015.pdf (it appears right in the title)

I also saw the first one \mathcal{H} in this question svmono template and `\mathcal`.

Answer 1

The H comes from the standard font cmsy10, but an old version. There was a change in 2009. (The cited typesetting-script.pdf was created at 2009-02-11, using the old symbol font.)

The symbol with cmsy10.pfb from an old TeX Live 5 (2000):

The font file cmsy10.pfb from TL 5 (2000) contains:

CMSY10 1.0
%%CreationDate: 1991 Aug 15 07:20:57
% Copyright (C) 1997 American Mathematical Society.  All Rights Reserved.

The current font file (2018) contains:

%%Title: CMSY10
%Version: 003.002
%%CreationDate: Mon Jul 13 16:17:00 2009
%%Creator: David M. Jones
%Copyright: Copyright (c) 1997, 2009 American Mathematical Society

Very likely there was a font change in 2009 or before that changed the calligraphic uppercase H.

Answer 2

I hacked a fit to 10pt and 12pt versions, by stacking white boxes over the left arm of the H.

\documentclass{article}
\usepackage{rotating,xcolor,stackengine}
\newcommand\specialHten{\mkern-3mu\rotatebox[origin=c]{-15.5}{%
\stackinset{l}{1.7pt}{c}{-.9pt}{\color{white}\rule{1.5pt}{1pt}}{%
  \stackinset{l}{1.875pt}{c}{-.81pt}{\color{white}\rule{1.5pt}{.65pt}}{%
    \rotatebox[origin=c]{15.5}{$\mathcal{H}$}}}}\mkern-3mu
}
\newcommand\specialHtwelve{\mkern-3mu\rotatebox[origin=c]{-15.5}{%
\stackinset{l}{1.9pt}{c}{-1.1pt}{\color{white}\rule{1.5pt}{1pt}}{%
  \stackinset{l}{2.53pt}{c}{-1pt}{\color{white}\rule{1.5pt}{.85pt}}{%
    \rotatebox[origin=c]{15.5}{$\mathcal{H}$}}}}\mkern-3mu
}
\begin{document}
$\mathcal{H}\specialHten\mathcal{H}$

\fontsize{12pt}{12pt}\selectfont
$\mathcal{H}\specialHtwelve\mathcal{H}$

\end{document}

If those boxes were visible, they would look like this:

Answer 3

Here is a version that gets you something similar using the modern toolchain (unicode-math in XeLaTeX or LuaLaTeX). A second script alphabet is available in Asana Math as a stylistic alternative.

\documentclass[varwidth, preview]{standalone}
\usepackage{unicode-math}

\setmainfont{Palatino Linotype}[Ligatures={Common,Discretionary,TeX}]
\setmathfont[Scale=MatchUppercase]{Asana Math}
\setmathfont[range={cal, bfcal}, Scale=MatchUppercase, Alternate]{Asana Math}

\begin{document}
\( \mathcal{H} \)
\end{document}

You can see a pretty comprenehsive list of the legacy calligraphic alphabets in the documentation for mathalpha (formerly mathalfa).

You might try pxtx or see if there’s another version there you like better:

\documentclass[varwidth, preview]{standalone}
\usepackage[cal=pxtx]{mathalfa}

\begin{document}
\( \mathcal{H} \)
\end{document}

Both of these are based on pxfonts/txfonts by Young Ryu.