6984
/www/nrich/html/content/id/6984/
1
2011-02-01T00:00:01
<mdoxml version="1.0"><br></br>
<span style="font-style: italic;">This problem is inspired by a
picture created by the artist Bridget Riley entitled "Movement in
Squares" (1961)</span><br></br>
<br></br>
Take two pieces of squared paper and colour alternate squares,
giving a checkerboard pattern.<br></br>
<br></br>
<div style="text-align: center;"><mdo:image width="400" height="295" alt="checkerboard pattern" src="flat.jpg"></mdo:image></div>
Now, curve your pieces round to create two cylinders, and stand
them up together.<br></br>
<div style="text-align: center;"><mdo:image width="400" height="300" src="cylinders.jpg" alt="cylinders"></mdo:image></div>
<br></br>
When you look straight at the two cylinders, the squares appear as
rectangles getting narrower and narrower as the page curves away
from you:<br></br>
<div style="text-align: center;"><mdo:image width="400" height="383" src="curved2.jpg" alt="closeup of cylinders"></mdo:image></div>
<br></br>
<h4>How could you represent this effect on a flat piece of
paper?</h4>
<br></br>
<div>The diagram below shows what you might see if you looked at
the cylinder from above. The construction lines may help you to
work out how to recreate the image of the cylinders in two
dimensions.</div>
<br></br>
<div style="text-align: center;"><mdo:image width="300" height="436" alt="bird's eye view" src="diagram.png"></mdo:image></div>
<div>You could vary the effect of your final image by altering the
size of the black and white squares.</div>
<br></br>
<div>We look forward to seeing your calculations and finished
products!</div>
<br></br>
<br></br></mdoxml>
<mdoxml version="1.0"><br></br>
<p><span class="editorial">Robert from Bishop Tonnos High School in Canada sent us the following solution:</span></p>
<p>In the picture viewing the cylinders from above, take each line that forms a radius of the circle, and draw a horizontal line between the left edge of the page and its contact point on the circle. The radial divisions of the circle are equal; we'll call this angle $\theta$.</p>
<p>Radius of the circle is $R$, and square size of the $n^{th}$ square will be denoted $S_n$.</p>
<p>It's clear that for the first square, $S_1 = R\sin\theta$.</p>
<p>For the second square, $S_2 = R\sin(2\theta)-R\sin\theta$</p>
<p>What this problem reduces to is finding the difference between horizontal lines, since it is the "overhang" which will determine the apparent size of a square. The difference between lines (and therefore the width of each square) is:</p>
<p>$$S_n=R\sin(n\theta)-R\sin((n-1)\theta)$$</p>
<p> </p>
<p><span class="editorial">Louis from Eltham College considered what would happen with a cylinder with $n$ divisions around it. You can read his solution <a href="/content/id/6984/Louis%20solution.pdf">here.</a></span><br></br>
</p></mdoxml>
<mdoxml version="1.0"><h3>Why do this problem?</h3>
<div>This problem gives a rather unusual take on two-dimensional
representations of three-dimensional shapes. It is based on the
artwork of Bridget Riley, so could provide an excellent
opportunity for forging cross-curricular links with Art and Design
departments.</div>
<br></br>
<div>Recreating the three-dimensional object as a two-dimensional
image provides a totally non-routine application of trigonometrical
calculations in a very creative context.</div>
<br></br>
<h3>Possible approach</h3>
<div>If images of Bridget Riley's work can be found, a nice
starting point would be to share these with the class, particularly
"Movement in Squares" (1961) which provided the inspiration for
this task.</div>
<br></br>
<div>Explain that the task is to create in two dimensions an image
that appears to show two cylinders, like the photo in the problem.
Give learners time to discuss in pairs or small groups ways that
they could do this, and then share these as a class. The image at
the bottom of the problem could be used to prompt a trigonometrical
or scale drawing approach.</div>
<div></div><br></br>
Once learners have devised a way to work out the measurements
needed to create the image, allow plenty of time to actually make
the images - an excellent opportunity for a classroom
display!<br></br>
<h3>Key questions</h3>
<div>How do the squares appear to change as the paper curves
away?</div>
<div>How can we calculate the changing widths of the squares?</div>
<h3>Possible extension</h3>
<div>Investigate creating images based on other curves, such
as a sphere, or a parabola. </div>
Do you think Bridget Riley's "Movement in Squares" is based on
cylinders? Make sure you can support your answer mathematically!
<br></br>
<h3>Possible support</h3>
Offer learners the image showing the view from above the cylinder
at an early stage.<br></br>
<br></br>
<br></br></mdoxml>
<mdoxml version="1.0">If this shows a quarter of the cylinder from above, what do you
know about the angles?<br></br>
What is the significance of the black and white sections at the
bottom of the image?<br></br>
<mdo:image width="300" height="436" src="diagram.png" alt="bird's eye view of cylinder"></mdo:image><br></br></mdoxml>
2
4
0
0
0
1
0
Moving Squares
How can you represent the curvature of a cylinder on a flat piece of paper?
Applications
Art
Trigonometry
Sine
3D Geometry, Shape and Space
2D representations of 3D shapes
2D Geometry, Shape and Space
Straight edge & compass constructions