This is what happened behind the scene.

In the first draft, I was very pedantic. The resulted draft is notation-heavy and has no introductory section. It was then totally discarded. However, the structure of the proof was reused in the later drafts. My advisor suggested a useful dictionary by Trzeciak for mathematicians like me who write in English as a second language.

The second draft was finished after 3 months. This time, I started with the history of the problem. The entire process of tracking down the history is like this.

The picture is exaggerated. The reality is that for the first time I was handling 10+ references at the same time and that I had to figure out how and in what order I should introduce them in my article. The introductory took 1/3 of the time. Another 1/3 of the time, I was struggling how I could turn the notation-heavy proof in my first draft into English words and sentences which are more friendly to readers. The rest 1/3 of the time, I was drawing pictures and figures. I had never used Tikz (a standard LaTeX graphics package) before. After I learnt how to draw points, lines, how to fill in shape with color and how to put labels, I started my ambitious project to draw a 3D triangulation of a sphere. The result was a super slow Ruby script that outputs Tikz code, which was later complied to generate pictures.

The hope was to have the top notch graphics in my paper to compensate the weak mathematics result I got.

Advisor was pretty happy with the second draft and provided several suggestions. Takeaways:

- In LaTex,
`:`

is a binary operator. The correct unary operator is`\colon`

. Use`\smallskip`

,`\midskip or`

`\bigskip`

to manually adjust spacing. - Citation information on MathSciNet is generally better than Google Scholar in terms of accuracy of information.
- Starting with the introduction section might be a better way to start a paper. By looking at others’ writings, one can familiarize himself/herself with the language and writing style of the field.

Advisor also drew me the following picture on my second draft.

The third draft was a revision of the second draft with two new figures. One of the major pictures is enhance by the Phong shading algorithm. A more refined fourth draft followed with more than 70 places of modifications.

The outcome of this creative process can be found on arXiv:1405.2503.