Is citing research publications in a blog post enough, or are there additional rights one must obtain?

Question

Say someone is writing a blog post on some science topic, and wants to show some mathematical expressions in the piece. What are the rights one has in showing equations that appear in published articles?

I imagine if the equation is a well-established, oft-used equation that nobody owns the expression/content (e.g. the equation for entropy or mutual information).

But if the equation is the creation of the author of some publication, as in the author is expressing some unique mathematical expression, obviously the publication should be referenced. But is referencing enough? Are there additional rights one must obtain from the publication and/or author in order to show this content?

For example, in theses one must obtain the rights to reference articles in addition to providing the citations.

One can imagine how arduous this would be (obtaining rights) for frequently published blog posts.

And how would this differ between say ArXiv which is open access, and a publication that is pay-only, like Elsevier or Nature?

Answer 1

how would this differ between say ArXiv which is open access, and a publication that is pay-only, like Elsevier or Nature?

It wouldn't.

But if the equation is the creation of the author of some publication, as in the author is expressing some unique mathematical expression, obviously the publication should be referenced. But is referencing enough? Are there additional rights one must obtain from the publication and/or author in order to show this content?

Laws of nature, including purported laws of nature, are not protected by copyright. So, usually, key equations in an academic paper aren't protected by copyright law.

Referencing the work is important as a matter of academic ethics, but is legally irrelevant.

Copying of exposition by the author beyond laws of nature is permitted as fair use if it is limited to quotations necessary for academic discussion and criticism, but copying of an entire work would not be permissible fair use in most cases and would constitute copyright infringement. Of course, there is a large gray area between those extremes.

Answer 2

After researching, and talking to publishers, this seems like the most appropriate approach:

While nobody can own math, or the underlying physical/informational principles, the exact expression and flow of arguments is considered "content" that requires special copyright permission to be obtained for use.

For example, Elsevier requires one to obtain permission for using content via www.Copyright.com. This includes using a mathematical expression from one of their papers.

arXiv does not force authors to transfer any copyright to them. Instead, only an irrevocable right to redistribute the content is obtained. However, depending on the specific article, there may have been subsequent copyright transfer, in the case of published articles, or the article may assert a license that is incompatible with commercial publication.

So even for arXiv, they request that one contacts the copyright holder (author/publisher) for additional information, and specifics related to the work.

So, I believe the most prudent approach when doing your own technical writing (e.g. blogpost, textbook, etc.) is don't copy anything verbatim, including mathematical equations/formulae. Since there is no exact replication of content this would not constitute any kind of plagiarism or copyright violation. Again, nobody can own math or physical principles in general.

If your writing is inspired by a given paper, obviously cite the work. But recast the mathematics in your own fashion (your own specific arrangement of symbols). Obviously the use and structure of the math you present in your writing should differ in argument/flow than where the inspired math came from.

If you must reuse the actual content (exactly same equation and/or flow of arguments) then both cite the original work and obtain the additional permissions from the publisher or author.

I am not a lawyer, so I would be happy to hear the opinions of those more knowledgable in this area.