Question:

Why can't we write the entire 24 volumes of the Encyclopedia Britannica on the head of a pin?

Lucas: 3 days ago

In his 1959 talk titled There's Plenty of Room at the Bottom (https://en.wikipedia.org/wiki/There's_Plenty_of_Room_at_the_Bottom) Feynmann discussed the miniaturisation of information storage and as part of this proposed that the entire contents of the Encyclopaedia Brittanica could be written on a pin head. A PDF of the talk is available here (http://calteches.library.caltech.edu/47/2/1960Bottom.pdf).

However I don't think we can input their information on $1 mm^2$. Is it impossible to create many information on $1 mm^2$ in real? Actually now there isn't such a thing after 50 years later. If it's possible, please tell me that mechanism.

Answer:
Cooper: 3 days ago

A DNA strand (https://en.wikipedia.org/wiki/DNA) has a diameter of about $2\times 10^{-9}\,\mathrm{m}$, and let's assume we can encode a bit in a length of $2\times 10^{-9}\,\mathrm{m}$ of DNA. Further assume we can lay DNA strands next to each other with no space between them.

With these assumptions a bit requires $4\times 10^{-18}\,\mathrm{m}^2$. The area of a pinhead is approximately $10^{-6}\,\mathrm{m}^2$, so you can fit about $2.5\times 10^{11}$ bits on a pinhead this way.

The Encyclopaedia Britannica seems to fit on a DVD (at least you can buy it on a DVD according to Wikipedia). The biggest DVDs (DVD-18 (https://en.wikipedia.org/wiki/DVD)) seem to be about $16\times 2^{30}$ bytes, which is about $1.7\times 10^{11}$ bytes or about $1.4\times 10^{12}$ bits.

So, encoding data like this we are short by a factor of about ten. However I have been fairly pessimistic about the encoding density of DNA: a nucleotide, which is two bits, seems to take only about $3.3\times 10^{-10}\,\mathrm{m}$ in fact (see above reference). Also I have overestimated the size of the Encyclopedia Britannica: according to this (https://en.wikipedia.org/wiki/Encyclop%C3%A6dia_Britannica), the last printed edition was $32,640$ pages, which, assuming $100$ lines of $100$ characters per page (an overestimate), one byte per character (a slight underestimate perhaps), is about $2.6\times 10^9$ bits, which will fit easily even including illustrations.

I have ignored compressibility of the data and the need for redundancy in the representation: assume they cancel out.