(Another variation of an IMTS problem)

"*It doesn't matter how beautiful your theory is, it doesn't matter how smart you are. If it doesn't agree with experiment, it's wrong.*"

"*The thing that doesn't fit is the thing that's the most interesting: the part that doesn't go according to what you expected.*"

- both quotes by Richard R. Feynman (1918-1988), famous American theoretical physicist & joint Nobel Prize in Physics winner in 1965

**Problem**

**Note**: the following curious, paradoxical geometry problem is yet another variation of an International Mathematical Talent Search (IMTS) problem available CLICK HERE.

"Determine the area of the **crossed** octagon *ALBMCNDO* as a fraction of the area of the square *ABCD*, where *L*, *M*, *N*, *O* are the midpoints of the sides of the square as indicated."

**Explore**

1) Click on the '**Show Areas**' button below to view the two areas concerned. Drag any of the red vertices to explore.

2) What do you notice about the area of the crossed octagon *ALBMCNDO* in relation to that of the square? Do you find it surprising? Or perhaps even contradictory or confusing?

a) What is happening here? The situation looks quite paradoxical!

b) So the question is **WHY** does the software show the area of the crossed octagon equal to that of the square while the total area shaded in green is clearly less than that of the square?

c) Can the situation be explained & clarified?

3) Also note that the formed crossed octagon is not regular. However, it has all sides equal and alternate angles are equal as can be seen by clicking on the '**Show Side Lengths**' and '**Show Angles**' buttons. This crossed octagon is therefore what I've called a semi-regular side-gon, specifically in this case, a (crossed) '*semi-regular side-octagon*' - which is a generalization to 2*n*-gons of the concept of a rhombus.

Area of crossed semi-regular side-octagon

**Challenge**

4) **Paradox**: Why is the area of the crossed octagon *ALBMCNDO* as calculated/determined by the software NOT equal to the total area of all the parts shown & shaded in '**green**'by the software?

5) In order to correctly determine the area of a crossed polygon one has to use 'signed areas' as explained by Coxeter & Greitzer (1967). Using signed areas, can you explain why (prove that) the area of the crossed octagon *ALBMCNDO* equals the area of square *ABCD* as observed above? Can you explain (prove) it in more than one way?

Click on the '

6) But why does the shaded representation in green of the crossed octagon

7) To further explain why it is a misleading representation, click on the '

8) To continue in the same fashion as described in 7) above, click on the '

9) Click on the '

10) What the above therefore shows is that while the area measurement of the area of the crossed octagon

Other dynamic geometry software have similar issues when dealing with representing crossed polygons, though all of them correctly measure/determine the area of the crossed octagon above. GeoGebra, for example, gives exactly the same representation as Sketchpad for the crossed octagon

However, Cinderella as shown in the second picture below, provides a more accurate representation than the other three software of the crossed octagon by showing the shading without the eight 'holes'. But it still does not show the number of overlays of the various triangles (as in steps 8) and 9) above).

**Submitted Paper**

A paper on this topic entitled "A Geometry Puzzler and its Explanation" will in due course be submitted to the *Learning & Teaching Mathematics* journal.

**References**

Coxeter, H. S. M. & Greitzer, S. L. (1967). *Geometry Revisited*. MAA, Washington, pp. 51-54.

De Villiers, M. (1994/2009). Some Adventures in Euclidean Geometry. Morrisville, NC: Lulu Publishers.

De Villiers, M. (1998). A Sketchpad discovery involving triangles and quadrilaterals. *KZN Mathematics Journal*, 3(1), 18-21.

De Villiers, M. (2014). Slaying a Geometrical Monster: Finding the Area of a Crossed Quadrilateral. *Scottish Mathematical Council Journal*, 44 (Dec), pp. 71-74.

De Villiers, M. (2015). I have a dream: Crossed Quadrilaterals - a Missed Lakatosian Opportunity?. *Philosophy of Mathematics Education Journal*, No. 29, July.

De Villiers, M. (2020). The Value of using Signed Quantities in Geometry. *Learning and Teaching Mathematics*, No. 29, pp. 30-34.

**Related Links**

International Mathematical Talent Search (IMTS) Problem Generalized

Another parallelogram area ratio

Area Formula for Quadrilateral in terms of its Diagonals

Area ratios of some polygons inscribed in quadrilaterals and triangles

An Area Preserving Transformation: Shearing

Sylvie's Theorem

Some Parallelo-hexagon Area Ratios

Area Parallelogram Partition Theorem: Another Example of the Discovery Function of Proof

Finding the Area of a Crossed Quadrilateral

Crossed Quadrilateral Properties

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Created by Michael de Villiers with *WebSketchpad*, 9 Oct 2023; updated 4 March 2024.