Archimedes’ parabolic area formula for cubics!

I try to post a new mathematics video once a week, either at my original YouTube site Insights into Mathematics, or my sister channel Wild Egg mathematics courses. This weekend’s post is particularly interesting I think, because it represents also the first “publication” of this material, albeit in an unusual format –YouTube instead of a paper in an established mathematics journal.

Here is the video that presents this new result, at Wild Egg mathematics courses. The video description contains the following:

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The very first and arguably most important calculation in Calculus was Archimedes’ determination of the slice area of a parabola in terms of the area of a suitably inscribed triangle, involving the ratio 4/3. Remarkably, Archimedes’ formula extends to the cubic case once we identify the right class of cubic curves. These are the de Casteljau Bezier cubic curves with an additional Archimedean property, characterized either by the nature of the point at infinity on the curve, or alternatively by the geometry of the quadrilateral of control points.

This is a very pleasant situation, and shows the power of the Algebraic Calculus to not only explain current theories more carefully and correctly, but also to discover novel results and open new directions.

I should have mentioned in the video that this Archimedean situation covers also the special case of a cubic function of one variable, that is a curve with equation y=a+bx+cx^2+dx^3.

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Posting research directly to YouTube, or some other web place, is quite an important development I believe. Here I am foregoing the usual refereeing process and uploading the material to the world, or in practice anyone interested in it who can find it. Should academics be allowed to do this?

On the one hand the work has not been peer reviewed, but these days peer review is often problematic, with most papers in pure mathematics almost certainly not being reviewed carefully and critically. This is not due to laziness or negligence, rather it is a necessary consequence of the increasing specialization and complexity of the subject. Most reviewers do not have the several weeks, or months, that it would typically require to delve into the details of a longish and complicated paper. It is understandable that on average they only skim the results and try to selectively check accessible proofs.

On the other hand, this new process completely sidesteps the usual gatekeepers of knowledge, namely editors and referees. Journals are often oriented to certain points of view or orthodoxies, unstated yet omnipresent. Perhaps they are entering a new phase when they will have to share relevance with the wiki processes of people deciding directly which content creators they value and trust.

In the meantime, I hope you enjoy the idea of a two thousand three hundred year old calculus result being extended to the next level!

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