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As part of my secondary geometry class I like to hook students by presenting real-world examples (usually images I find online or have taken myself) of different geometric shapes from real life. For instance, a lesson on the area of a circle might start out with a picture of a pizza pie or a lesson on the midsegments of triangles might start out with a picture of the Triforce. However, there are some geometric figures that I have had a hard time finding interesting, real-world examples of. Those figures (and I know I am forgetting a bunch..) are:

  • Segment of a circle
  • Secant line
  • Trapezoid (Isosceles or not)
  • Inscribed angle
  • Parallel lines cut by a transversal

I was wondering if anyone had any ideas for these geometric figures of interesting, real world examples? Also, I think it would be great that if people are aware of really cool real world examples for the more "standard" geometric figures to post those as answers as well. For instance, the Dockland Building at the Port of Hamburg is an astoundingly perfect parallelogram :) enter image description here Having a collection would be very helpful for teachers because I have not found a better way to get my students right into the groove by starting class off with a brief discussion about an interesting picture!

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    $\begingroup$ For some of these, start by closing your eyes and imagining where in life you have seen such shapes. I can imagine leaning books on a bookshelf and train tracks intercepting a road at an angle for your trapezoid and for parallel lines intercepted by a nonperpendicular line. Doubtless a search of an image database will come up with other examples present in the world. Gerhard "You Can Ask Your Students" Paseman, 2015.03.05 $\endgroup$ Mar 5, 2015 at 17:33
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    $\begingroup$ There is an MO question on mathematical sculptures. Mostly they seem to ornate for the present purpose, but some might be relevant. $\endgroup$
    – quid
    Mar 6, 2015 at 12:09
  • $\begingroup$ Quadrilaters: there's also kites (normal kite). And concave kites (also actual kites). Also the Star Trek insignia. And arrowheads. $\endgroup$
    – guest
    Jun 20, 2018 at 21:57

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Hexagonal basalt columns at the Giant's Causeway in Northern Ireland:


          HexCols
          (Image from Wikipedia.)
         
          (Image from RTomlinson.)


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Ever notice the $5$-point star at the base of a pumpkin stem? This one's pentagonal symmetry is especially evident:


          Pump_both
          Two views of the same pumpkin.


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From an MathOverflow question, Six yolks in a bowl: Why not optimal circle packing?:


         
          Six yolks in a bowl.

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Parabolas in a fountain at Parque das Águas, Cuiabá, Brazil.

enter image description here

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Synergia, built with congruent space-filling hendecahedra: https://exhibitcolumbus.org/exhibition/synergia

ref: https://www.jstor.org/stable/3618509

… and a unremarkable regular icosahedron in a playground near my neighborhood: enter image description here

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"Two Swedish architects named Bigert & Bergstrom showcased their newest creation… the Solar Egg. This 16-foot tall egg-shaped structure is actually a sauna which burns wood to create heat."


          SolarEgg
          Image from here.
The tesselation appears to be irregular.

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As Wikipedia notes, Euler spirals, also known as clothoids, are

widely used in railway and highway engineering to design transition curves between straight and curved sections of railway or roads

An application that may be more relatable to students is in the design of rollercoaster loops, as pointed out in this post (with a couple of pictures) at ThatsMath.

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Many molecules have striking shapes. For example cubane: https://en.wikipedia.org/wiki/Cubane

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    $\begingroup$ Short and link-only answers are discouraged. A summary of the contents of the link would improve the answer. $\endgroup$
    – Tommi
    Jan 22, 2019 at 12:17
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An interesting curve I recently noticed in some reading is the Hawaiian earring which we can denote $\mathbb{H}$ for the purpose of this answer. $$ \mathbb{H} = \{ (x,y) \in \mathbb{R}^2 \ | \ (x-1/n)^2+y^2=1/n^2, n=1,2,\dots \}. $$ In other words, $\mathbb{H}$ is the union of all circles with center $(1/n,0)$ and radius $1/n$ where $n$ is a positive integer. Here is a partial picture of $\mathbb{H}$, it just has the the $10$ largest circles (found this in the Wikipedia article linked below) enter image description here There are also higher dimensional analogs of this construction where the circles are replaced by higher dimensional spheres. The topology on $\mathbb{H}$ is very interesting as you can read about here:https://en.wikipedia.org/wiki/Hawaiian_earring

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    $\begingroup$ I think his is not an answer to the question? $\endgroup$ Aug 11, 2023 at 12:57
  • $\begingroup$ @GeraldEdgar you're probably right, I'm answering the spirit rather than the letter of the question, and, maybe I have twisted the spirit into a different dimension. $\endgroup$ Aug 12, 2023 at 1:22
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