Bullseye illusion

Bullseye illusion: The inner circle appears larger than the outer ring

The Bullseye illusion is a geometric optical illusion.

Description

In Figures 1 to 4, the green circle appears larger than the blue ring. In reality, however, both areas have the same area.

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  Figure 1
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  Figure 2
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  Figure 3
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  Figure 4

Mathematical background

The blue ring is formed by the difference in area between a larger circle with radius c and a smaller circle with radius b.

Let the green circle have radius a. Then the condition for the equality of the two areas is:

${\displaystyle \pi a^{2}=\pi c^{2}-\pi b^{2}}$, i.e.,

${\displaystyle a={\sqrt {c^{2}-b^{2}}}}$.

For the special case ${\displaystyle a=b}$ (Figure 4), the condition for the equality of the two areas is:

${\displaystyle \pi b^{2}=\pi c^{2}-\pi b^{2}}$, i.e.,

${\displaystyle a=b={\frac {c}{\sqrt {2}}}}$.

In this case, the larger circle is completely decomposed into the smaller green circle and the blue ring of equal area. ^{[1] [2]}

External links

• Bullseye illusion Wolfram MathWorld
• Bullseye illusion (interactive demonstration) Wolfram Demonstrations Project
• Bullseye optical illusion from math-fail.com

References

1. Claudi Alsina, Roger B. Nelsen: Pearls of Mathematics - 20 Geometric Figures as Starting Points for Mathematical Explorations (english Original title: Charming proofs – A Journey into Elegant Mathematics), Springer-Verlag GmbH Berlin 2015, ISBN 978-3-662-45460-2, page 141
2. Wells, D.: The Penguin Dictionary of Curious and Interesting Geometry. Penguin Books, London (1991), page 87