Fibonacci numbers inside the Fibonacci word fractal. Counting the "gaps", at any scale, within the pattern always give Fibonacci numbers. That is also the case for the 2 other types :  <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{3k}}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><msub><mi>F</mi><mrow class="MJX-TeXAtom-ORD"><mn>3</mn><mi>k</mi></mrow></msub></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle F_{3k}}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89e27fe33ce9673974bb30938507667bfb841333" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:3.405ex; height:2.509ex;" alt="{\displaystyle F_{3k}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{3k+1}}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><msub><mi>F</mi><mrow class="MJX-TeXAtom-ORD"><mn>3</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle F_{3k+1}}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04e2272fd6bdbc5fba444cff9aaace4f50120eb6" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:5.506ex; height:2.509ex;" alt="{\displaystyle F_{3k+1}}"></span>. See also The "Fibonacci word gasket".
<a href="//commons.wikimedia.org/wiki/User:Prokofiev" title="User:Prokofiev">Prokofiev</a>
CC BY-SA 3.0
Own work by uploader <a rel="nofollow" class="external text" href="http://hal.archives-ouvertes.fr/hal-00367972/fr/">Reference</a>