Real-world examples of more "obscure" geometric figures

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 :) 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!

• 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 Commented Mar 5, 2015 at 17:33
• There is an MO question on mathematical sculptures. Mostly they seem to ornate for the present purpose, but some might be relevant.
– quid
Commented Mar 6, 2015 at 12:09
• Quadrilaters: there's also kites (normal kite). And concave kites (also actual kites). Also the Star Trek insignia. And arrowheads. Commented Jun 20, 2018 at 21:57

Trapezoid

Native Peruvian architecture makes heavy use of the trapezoid for stability in earthquakes. (The Spaniards thought they were primitive as they didn't use arches ... but most of the Spanish buildings have collapsed or had to be rebuilt).

It's especially apparent in their doorways and windows.

(hi res)

Other examples with licensing such that I don't want to embed it:

Trapezoids are also found in cabinet joinery, specifically dovetail joints.

Segment of a Circle

Most architectural arches are based on segments of circles, particularly those in Roman architecture:

(hi res)

(hi res)

Chinese architecture tends to favor segmental arches (which the Romans also used), rather than full rounded arches:

See also arch bridges and barrel vaults. If you want sophomoric humor, also consider the groin vault (when made with round barrel vaults, not pointed barrel vaults).

Parabola

There are also Parabolic Arches:

Parallel lines cut by a transversal

Runways at large airports. They typically have taxiway parallel to the runway, and in windy areas have a second (or even third) pair to avoid takeoffs/landings into a cross wind. BWI is a good example, but I was having difficulty finding public domain images. Here's one of O'Hare:

(hi res)

• That’s no m̶o̶o̶n̶ parabola. That’s inverted catenary. Commented Aug 19, 2015 at 19:12
• @IncnisMrsi : which one, the roof or the arch? (I'm not good with telling 'em apart by eye)
– Joe
Commented Aug 20, 2015 at 12:32
• The arch is catenary. The roof is likely parabolic, although it is not bent enough to determine its mathematical prototype (if any) reliably. Commented Aug 20, 2015 at 19:58
• @IncnisMrsi : Replacement parabolic arches : myarchitecturalvisits.com/2015/03/19/… . (found via cs.rutgers.edu/~mcgrew/dimacs/slides/Amadeo_Huylebrouck.pdf , which is a presentation on fitting of architectural arches)
– Joe
Commented Aug 21, 2015 at 15:17

I like the Gateway Arch in St. Louis as an example of a catenary with a formula of the form $y= A \cosh(\frac{C X}{L}) -A$. More information on the wiki: Gateway Arch: Mathematical Elements.

• It should be mentioned that it's much easier to find non-inverted catenaries, since power lines will take that shape. Commented Mar 7, 2015 at 3:56
• @Dietrich Epp … but on short runs between two poles it’s hard to distinguish a catenary from a parabola. Commented Aug 20, 2015 at 20:04

Dice

You get all Platonic solids, some trapecohedrons and bipyramids, and the tetrahexahedron and the rhombic triacontahedron:

The hexagon at the north pole of Saturn:

It is known that

"[regular shapes] form in an area of turbulent flow between ... two different rotating fluid bodies with dissimilar speeds."

and this has been proposed as an explanation for the phenomenon.

Incidentally, the Earth could easily fit inside the pole hexagon.

Added (23Sep15). An article in space.com cites a new and apparently thorough explanation of Saturn's polar hexagon, in The Astrophysical Journal Letters:

Here we present numerical simulations showing that instabilities in shallow jets can equilibrate as meanders closely resembling the observed morphology and phase speed of Saturn's northern Hexagon.

Added (10Dec16). New images taken by Cassini:

• Incidentally, the northpole hexagon has changed color in the last four years! See space.com for Casini color images. Commented Oct 27, 2016 at 16:00

There's a fair attempt at a Hypercube with the Grande Arche de la Défense in Paris.

• I don't think "in/out" counts as another dimensions though. Commented Apr 12, 2015 at 1:15
• @PyRulez Do you think you can draw a cube on a piece of paper? Presumably you do, since you seem happy that the outer structure in this photo is a cube. If that's ok, why do you object to projecting the fourth dimension into three? Commented Aug 24, 2015 at 12:50
• @JessicaB When I draw a "cube", I'm only drawing a representation, not an actual cube. Likewise, they didn't build an actual hypercube, just a representation. Its even still a representation in real life, not just the photo. Saying this is a actually hypercube would be like saying dodecahedron in a movie are actual dodecahedron. Commented Aug 24, 2015 at 13:42

A corkscrew (for a helix):

A donut (for a torus):

A football (for a spheroid)

And then, there's also the atomium (for which I am not sure exists a geometric name)

cooling towers (for a hyperboloid)

and the pentagon (well, for a pentagon):

A pyramid is, of course, a pyramid.

Lastly, a soccer ball is a truncated icosahedron

(Images by wiki, pedia)

• I'd +1 if you listed which geometric shapes these are examples of. (Well, OK, the Pentagon is kind of obvious.) For example, cooling towers are typically hyperboloids. Commented Mar 6, 2015 at 22:46

"Turning Torso," an apartment building in Malmö, Sweden designed by architect Santiago Calatrava, following a twisting spiral. It consists of "nine segments of five-story pentagons that twist relative to each other as it rises; the topmost segment is twisted 90 degrees clockwise with respect to the ground floor."

Sphere or hemisphere: Pantheon

One other poster mentioned arches; I'd like to add in the Gothic arch as an example of circular segments. These are great examples of arcs as well. I find them much more interesting, and they don't always have to have the angle shown here; the location of the circle's center can vary depending on the "slope" of the arch that is desired. There are also three- and four-centered arches. I can imagine that you could differentiate for your more advanced students by having them try to figure out how the more complicated arch structures were designed. Calculations related to the complicated structures could be somewhat intense but a fun challenge for a gifted student. The area underneath one of the simpler arches would be an interesting problem more at the level of the majority of the class.

Really great answers! I just found this while making a lesson on annuli, an Annular Eclipse, very beautiful! and it also has interesting math behind why the sun is not entirely covered by the moon!

At Wikipedia: Saddle roof you can see images of rooftops which are a Hyperbolic Paraboloid. Other "saddle-like" objects may also be this shape - the primary advantage of which (like its cousin the one-sheet hyperboloid i.e. nuclear plant cooling tower) is that it can be formed from supports that are straight lines in a grid.

At Hyperboloid structure you can see some radio towers that use the one-sheet hyperboloid as their shape.

• The Mae West in Munich is another example of a hyperboloid. Commented Mar 9, 2015 at 13:35

As constrast to the catenary in Chris's answer, you could show a suspension bridge, which has a parabola...

According to LINK, the curve in a suspension bridge is generally a curve intermediate between a catenary and a parabola.

• A parabola is the approximation where the weight of the cables is 0, so only the weight of the horizontal bridge deck counts. A catenary is the "approximation" where the weight of the bridge deck is zero, so only the weight of the cables count. The latter is an absurd approximation for a bridge, but it's accurate for a chain hanging all by itself. Commented Mar 6, 2015 at 3:18
• P.S. Years ago, in the early days of pocket calculators, one of the relevant companies (I forgot whether it was HP or TI) put a two-page ad in Scientific American, showing a picture of a suspension bridge under the equation of a catenary. Commented Mar 6, 2015 at 3:20
• Does the weight of the vertical cables also have to be 0 for it to be either one of these? Commented Mar 9, 2015 at 12:49
• See the LINK in the added comment. Cables weight zero -> parabola; bridge floor weight zero -> catenary. Commented Mar 9, 2015 at 13:26
• @GeraldEdgar My question is about the vertical cables having significant weight. The main cable alone should be a catenary - when the higher parts of it have longer vertical cables hanging from it than the shorter ones it should obviously be different. Commented Mar 9, 2015 at 13:51

A (cata)caustic is the envelope of lines reflected in a curve. The caustic formed by parallels lines reflected in a semicircle is a cardioid, such as can be seen in the bottom of this MSE coffee mug.

Other envelopes include evolutes. An evolute is the envelope of the normal lines to a given curve; the given curve is the involute of the evolute.

A famous involute is the cycloid, which the involute of itself (and therefore the evolute of itself, too). Because the cycloid is a tautochrone, Huygens used it to design a clock (left, Fig. II), which Coster made (right):

The involute of a circle (the smaller ones) can be used to design gear teeth that roll off each other without slipping (thus minimizing heating due to friction):

(Inspired by Gerhard's comment) Trapezoid:

(Image from Parth Chandran @emaze.com.)

• One could also consider the entire shape to be a frustum of a square pyramid. Commented Apr 17, 2019 at 3:06

The stone spheres (or stone balls) of Costa Rica are an assortment of over three hundred petrospheres in Costa Rica, located on the Diquís Delta and on Isla del Caño. Locally, they are known as Las Bolas (literally The Balls). The spheres are commonly attributed to the extinct Diquís culture and are sometimes referred to as the Diquís Spheres.

The Palmar Sur Archeological Excavations are a series of excavations of a site located in the southern portion of Costa Rica, known as the Diquís Delta. The excavations have centered on a site known as "Farm 6", dating back to the Aguas Buenas Period (300-800AD) and Chiriquí Period (800-1550 AD).

They are almost perfectly round, developed by a culture without any knowledge on geometry?

For a super-ellipse, one example would be the fountain at Sergels torg, in Stockholm, Sweden.

For a circular segment, one example would be the cross-section of liquid in a horizontal-axis circular cylinder tank. (Another picture is here.)

So called tensile structures in architectures are indeed minimal surfaces. Popular examples are

• the Olympiastadium in Munich: or
• the former Millenium Dome in London:

An ellipse as a cylindric section: The top surface of the Tycho Brahe Planetariun, Copenhagen, Denmark.

The building itself is a cylindrical segment.

Reggio Emilia Calatrava's railway station follows some very interesting geometrical patterns, building pairs of sinusoids in phase and out of phase

The Mito Art Tower consists of $$28$$ congruent, stacked regular tetrahedra, each with edge length about $$10$$m. It is in Mito, Ibaraki, Japan. Architect: Arata Isozaki.

Left image from [www.panoramio.com](http://www.panoramio.com/).
Right figure from Elgersma & Wagon. "The Quadrahelix: A Nearly Perfect Loop of Tetrahedra." 2016. [arXiv abstract](https://arxiv.org/abs/1610.00280).

Known as the Boerdjik-Coxeter helix.

Water towers:

The form comes from the need to (approximately) maintain a constant pressure.

I've found that students are not very clear on the image that is being invoked when I call $$z = x^2-y^2$$ by its traditional name of "saddle point", but they are all very clear on what a Pringles potato chip looks like.

• Don’t miss the chance to contrast this with the (knock-off) Stax chip by Lays, which is something like a parabolic cylinder. Commented Nov 9, 2020 at 13:19

The Puerta de Europa (Gate of Europe) in Madrid consists of two $$26$$-floor prisms inclined $$15^\circ$$, a nice example of a spatial shear transformation:

Designed by architects Senteri Cardillo, Philip Johnson and John Burgee.

Minimal surfaces was mentioned. Another example of minimal surfaces is soap bubbles:

• A convex surface is minimal? RoTFL. One might have no sane idea on physics of a membrane with some gauge pressure on it (such a soap bubble is) to claim it’s minimal. Commented Aug 20, 2015 at 20:22
• @Incnis Mrsi: Wikipedia here: en.wikipedia.org/wiki/Soap_bubble seems to disagree. What is being minimized is volume. Commented Aug 21, 2015 at 16:14
• Wikipedia has a lot of knowledgeable guys, but it is also notorious for deeply entrenched culture of irresponsibility. Here you can read how one William M. Connolley pointed out the mistake in 2007, although local incompetent text-writers either ignored or tried to debunk his criticism. Find a physics student around and ask him/her. A minimal surface, by definition, minimizes area (locally), not volume. Commented Aug 21, 2015 at 19:48
• Soap bubble minimize area given the enclosed volume, and are not minimal surfaces (but have constant, non zero mean curvature). Soap films (locally) minimize area given their boundary but are usually not considered minimal surfaces because of their singularities. Last, beware that in math there is a subtle difference between minimal surfaces and area minimizing surfaces (the former being a more general notion). Commented Aug 21, 2015 at 21:05

Spiral = snail shell.

Broccoli = fractal

-or- broccoli = decision tree (but a tree can also be a decision tree). Note that the slang term in the Navy for broccoli is "trees" (like sliders for burgers).

Wankel engine rotor has a similar curve-y triangle shape to the criticized coin above.

Drill chuck = truncated cone (also some of the internals of an automotive differential)

"Stadiums" for trapezoidal cylindrical shells (calculus volume of rotation problems)

Lots of other cool gear-type shapes (prop screw for a ship, pump lobes, camshaft, chevronic separators in boilers, tricone rotary drill bit). Not a 100% sure what they all correspond to math-name wise, but they definitely engage some wonder about shape to function.

A nice challenge for a calculus class with a little physics: If particles are thrown out from a common point in all directions at the same speed, then allowed to fall freely, the shape they will sweep out is a parabola. (Of course, the trajectory of each particle is also a parabola, that's a simpler fact.) The Fourth of July might suggest some examples:

When I was in high school, I saw a cutting board lying on an angle in a sink with the water pouring from the faucet onto a point on it. The water splashed out to form a parabolic arc. I wonder if you could actually bring something like that into the classroom and trace the edge of the water?

A just-released image of a crudely—but recognizably hexagonal—crater (the "Haulani Crater") on the dwarf planet Ceres (between Mars & Jupiter), taken by the Dawn spacecraft.

One article says it "look[s] oddly like a stop sign," but we know stop signs (in the U.S.) are octagons. How a physical process (asteroid collision) could result in an approximate hexagon is (I think?) not yet understood.

Cf. Saturn's north-pole hexagon, which is better understood (at least conjecturally).

Curves of constant width, the simplest of which is the Reuleaux Triangle, occur in a variety of applications. As a shape it consists of pieces of three circles. To construct a Reuleaux Triangle start with an equilateral triangle of side length h and with a compass from each vertex draw a circular arc with radius h between the other two vertices. The resulting set, like a circle has constant width h. Read more about the Reuleaux Triangle and its interesting properties here:

https://en.wikipedia.org/wiki/Reuleaux_triangle

(Image from de.ucoin.net.)

• I'm not sure if this counts as a "real-world example."
– JRN
Commented Jun 20, 2018 at 5:20
• @JoelReyesNoche, real world examples of curves of constant width such as the Reuleaux triangle would be some British coins or the internals of a Wankel engine. Commented Jun 20, 2018 at 7:27
• @PeterTaylor: Nice coin example. I took the liberty of adding an image. Commented Jun 20, 2018 at 10:02
• See my followup: Why are some coins Reuleaux triangles?. Commented Jun 20, 2018 at 14:17

A square antiprism at One World Trade Center