## Real-world examples of more "obscure" geometric figures

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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 :) 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!

2For 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, 2015.03.05Gerhard Paseman 2015-03-05T17:33:52.820

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There is an MO question on mathematical sculptures. Mostly they seem to ornate for the present purpose, but some might be relevant.

quid 2015-03-06T12:09:13.190

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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)

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That’s no m̶o̶o̶n̶ parabola. That’s inverted catenary.

Incnis Mrsi 2015-08-19T19:12:02.943

@IncnisMrsi : which one, the roof or the arch? (I'm not good with telling 'em apart by eye)Joe 2015-08-20T12:32:33.600

The arch is catenary. The roof is likely parabolic, although it is not bent enough to determine its mathematical prototype (if any) reliably.Incnis Mrsi 2015-08-20T19:58:38.997

@IncnisMrsi : Replacement parabolic arches : http://myarchitecturalvisits.com/2015/03/19/col%C2%B7legi-de-les-teresianes/ . (found via http://www.cs.rutgers.edu/~mcgrew/dimacs/slides/Amadeo_Huylebrouck.pdf , which is a presentation on fitting of architectural arches)

Joe 2015-08-21T15:17:00.293

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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.

2It should be mentioned that it's much easier to find non-inverted catenaries, since power lines will take that shape.Dietrich Epp 2015-03-07T03:56:25.860

@Dietrich Epp … but on short runs between two poles it’s hard to distinguish a catenary from a parabola.Incnis Mrsi 2015-08-20T20:04:15.680

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# Dice

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

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

So cool! :-) ${}$Joseph O'Rourke 2015-03-06T18:44:55.087

1I don't think "in/out" counts as another dimensions though.PyRulez 2015-04-12T01:15:34.070

1@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?Jessica B 2015-08-24T12:50:19.613

@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.PyRulez 2015-08-24T13:42:25.260

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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)

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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.

Ilmari Karonen 2015-03-06T22:46:15.160

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Sphere or hemisphere: Pantheon

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"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."

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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.

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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.

Joseph O'Rourke 2016-10-27T16:00:31.333

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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!

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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.

nwellnhof 2015-03-09T13:35:28.263

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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.

11A 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.Andreas Blass 2015-03-06T03:18:31.590

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.Andreas Blass 2015-03-06T03:20:01.490

Does the weight of the vertical cables also have to be 0 for it to be either one of these?Random832 2015-03-09T12:49:56.457

See the LINK in the added comment. Cables weight zero -> parabola; bridge floor weight zero -> catenary.Gerald Edgar 2015-03-09T13:26:02.227

@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.Random832 2015-03-09T13:51:58.943

@Random832 I think that if the weight of the main cables is zero and the sum of the weights of the vertical cables and the deck is nonzero, then the shape will be a parabola. A segment of the main cable can't tell whether the downward force on it is coming from the weight of the vertical cable below it or the weight of the deck below it.Trevor Wilson 2015-03-12T03:40:13.887

Right, but the vertical cables have a different weight distribution. It's an academic question, in reality they're never going to weigh more than either the main cable or the bridge deck.Random832 2015-03-12T03:41:11.710

@Random832 Oh, you are right. (But I will let my comment stand in case anyone else makes the same mistake.)Trevor Wilson 2015-03-12T03:42:58.713

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(Inspired by Gerhard's comment) Trapezoid:

(Image from Parth Chandran @emaze.com.)

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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.)

The related super egg shape is also rather interesting: https://en.wikipedia.org/wiki/Superegg

Joseph Malkevitch 2017-03-09T15:31:26.187

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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?

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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):

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

The building itself is a cylindrical segment.

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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.
Right figure from Elgersma & Wagon. "The Quadrahelix: A Nearly Perfect Loop of Tetrahedra." 2016. arXiv abstract.

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

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

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

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Water towers:

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

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The Puerta de Europa (Gate of Europe) in Madrid consists of two $26$-floor prisms inclined $15^\circ$:

(Image from archiseek.com.)
Designed by architects Philip Johnson and John Burgee.

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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.Incnis Mrsi 2015-08-20T20:22:35.590

@Incnis Mrsi: Wikipedia here: https://en.wikipedia.org/wiki/Soap_bubble seems to disagree. What is being minimized is volume.

kjetil b halvorsen 2015-08-21T16:14:06.087

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.

Incnis Mrsi 2015-08-21T19:48:23.103

5Soap 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).Benoît Kloeckner 2015-08-21T21:05:25.430

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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).