# The geometry of the geometric series

**URL:** <https://discourse.julialang.org/t/the-geometry-of-the-geometric-series/131976>\
**Category:** Teaching & Outreach\
**Created:** [August 30, 2025, 7:34pm UTC](https://discourse.julialang.org/t/the-geometry-of-the-geometric-series/131976 "2025-08-30T19:34:41Z")\
**Posts on this page:** 7\
**Page:** 1

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**Author:** ![g2g](https://avatars.discourse-cdn.com/v4/letter/g/41988e/32.png) [@g2g](https://discourse.julialang.org/u/g2g)\
**Post date:** [August 30, 2025, 7:34pm UTC](https://discourse.julialang.org/t/the-geometry-of-the-geometric-series/131976/1 "2025-08-30T19:34:41Z")

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This year’s Summer of Math Exposition (SoME) judging starts this Wednesday (03 Sep 2025). This year’s version of the contest encourages expositions that would be helpful to high school math classes. I’m submitting [this essay](https://olarth.medium.com/the-geometry-of-geometric-series-ab594b229dc1) about how to visualize the geometric series sum as the area of a right triangle filled in with trapezoid slices, where each trapezoid area represents the value of a term in the geometric series.

I would like to also include a Julia and Makie generated animated GIF that animates the transformation of the series of overlapped similar triangles (whose areas represent the values of the individual terms in the geometric series) into a series of equivalent area non-overlapped trapezoids that fill in a right triangle. The animation would be helpful in visualizing the geometry of the geometric series but it would also be a good chance to show off the animation capabilities of Julia and Makie. However, while I was getting surprisingly helpful feedback from Gemini’s proof reading of the essay, Gemini said that it couldn’t access the full essay when I gave it the story link at [medium.com](http://medium.com). (So I had to keep copying and pasting the essay text into the Gemini prompt for it to give its thorough critiques.)

So that is my first question: can non-members of [medium.com](http://medium.com) read the full essay or are they limited to seeing just a sample of the essay’s beginning? (I tried to manage that story’s paywall settings, but the only control option for the essay was a cancel button. Do I have to become a member to allow non-members to read the full essay?)

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**Author:** ![eldee](https://avatars.discourse-cdn.com/v4/letter/e/b5a626/32.png) [@eldee](https://discourse.julialang.org/u/eldee)\
**Post date:** [August 31, 2025, 11:58am UTC](https://discourse.julialang.org/t/the-geometry-of-the-geometric-series/131976/2 "2025-08-31T11:58:53Z")

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> [@g2g](#):
>
> So that is my first question: can non-members of [medium.com](http://medium.com) read the full essay or are they limited to seeing just a sample of the essay’s beginning?

I can confirm that I can read the full essay as a non-member.

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**Author:** ![g2g](https://avatars.discourse-cdn.com/v4/letter/g/41988e/32.png) [@g2g](https://discourse.julialang.org/u/g2g)\
**Post date:** [August 31, 2025, 2:51pm UTC](https://discourse.julialang.org/t/the-geometry-of-the-geometric-series/131976/3 "2025-08-31T14:51:01Z")

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> [@eldee](#):
>
> I can confirm that I can read the full essay as a non-member.

Thanks. My guess is that Gemini is constrained from reading essays at URLs due to copyright concerns but apparently it is okay when the essay is copied as part of the prompt.

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**Author:** ![liuyxpp](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/liuyxpp/32/9870_2.png) [@liuyxpp](https://discourse.julialang.org/u/liuyxpp)\
**Post date:** [September 2, 2025, 5:36am UTC](https://discourse.julialang.org/t/the-geometry-of-the-geometric-series/131976/4 "2025-09-02T05:36:11Z")

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For sometimes ago I have used [medium.com](http://medium.com), non-members are restricted to a certain amount of articles. So nowadays, I just avoid reading articles from [medium.com](http://medium.com). I have so many bad experience with it.

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**Author:** ![g2g](https://avatars.discourse-cdn.com/v4/letter/g/41988e/32.png) [@g2g](https://discourse.julialang.org/u/g2g)\
**Post date:** [September 2, 2025, 11:32am UTC](https://discourse.julialang.org/t/the-geometry-of-the-geometric-series/131976/5 "2025-09-02T11:32:44Z")

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> [@liuyxpp](#):
>
> So nowadays, I just avoid reading articles from [medium.com](http://medium.com).

What about publishing … when you have an essay you want to freely distribute, what is your preference? (I ask because I’m not completely satisfied with [medium.com](http://medium.com) for that. For example, when Gemini was proofreading this essay, it suggested using LaTeX to improve the look of the equations and it even wrote a little javascript animation to show the transformation from overlapped similar triangle to non-overlapped trapezoid. Both were great suggestions but neither are allowed in the [medium.com](http://medium.com) editor.)

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**Author:** ![liuyxpp](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/liuyxpp/32/9870_2.png) [@liuyxpp](https://discourse.julialang.org/u/liuyxpp)\
**Post date:** [September 2, 2025, 11:34am UTC](https://discourse.julialang.org/t/the-geometry-of-the-geometric-series/131976/6 "2025-09-02T11:34:35Z")

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I write articles in my personal blog hosted by github.

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**Author:** ![Jake](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/jake/32/46007_2.png) [@Jake](https://discourse.julialang.org/u/Jake)\
**Post date:** [September 2, 2025, 1:34pm UTC](https://discourse.julialang.org/t/the-geometry-of-the-geometric-series/131976/7 "2025-09-02T13:34:21Z")

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> [@g2g](#):
>
> What about publishing … when you have an essay you want to freely distribute, what is your preference?

I see some people posting on SubStack and have been able to read the articles.