The Collatz map can be viewed as the restriction to the integers of the smooth real and complex map
<dl><dd><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(z)={\frac {1}{2}}z\cos ^{2}\left({\frac {\pi }{2}}z\right)+(3z+1)\sin ^{2}\left({\frac {\pi }{2}}z\right)}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mi>z</mi><msup><mi>cos</mi><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msup><mo>⁡<!-- ⁡ --></mo><mrow><mo>(</mo><mrow><mrow class="MJX-TeXAtom-ORD"><mfrac><mi>π<!-- π --></mi><mn>2</mn></mfrac></mrow><mi>z</mi></mrow><mo>)</mo></mrow><mo>+</mo><mo stretchy="false">(</mo><mn>3</mn><mi>z</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><msup><mi>sin</mi><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msup><mo>⁡<!-- ⁡ --></mo><mrow><mo>(</mo><mrow><mrow class="MJX-TeXAtom-ORD"><mfrac><mi>π<!-- π --></mi><mn>2</mn></mfrac></mrow><mi>z</mi></mrow><mo>)</mo></mrow></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle f(z)={\frac {1}{2}}z\cos ^{2}\left({\frac {\pi }{2}}z\right)+(3z+1)\sin ^{2}\left({\frac {\pi }{2}}z\right)}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e52c4ebde64874d076cbedc1d4a1518148b6402" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.838ex; width:42.177ex; height:5.176ex;" alt="{\displaystyle f(z)={\frac {1}{2}}z\cos ^{2}\left({\frac {\pi }{2}}z\right)+(3z+1)\sin ^{2}\left({\frac {\pi }{2}}z\right)}"></span>,</dd></dl><p>which simplifies to <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 {\frac {1}{4}}(2+7z-(2+5z)\cos(\pi z))}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow class="MJX-TeXAtom-ORD"><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow><mo stretchy="false">(</mo><mn>2</mn><mo>+</mo><mn>7</mn><mi>z</mi><mo>−<!-- − --></mo><mo stretchy="false">(</mo><mn>2</mn><mo>+</mo><mn>5</mn><mi>z</mi><mo stretchy="false">)</mo><mi>cos</mi><mo>⁡<!-- ⁡ --></mo><mo stretchy="false">(</mo><mi>π<!-- π --></mi><mi>z</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle {\frac {1}{4}}(2+7z-(2+5z)\cos(\pi z))}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f55a387c9d12c82ae998e689e40632961b7db47" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.838ex; width:28.692ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{4}}(2+7z-(2+5z)\cos(\pi z))}"></span>.
If the standard Collatz map defined above is optimized by replacing the relation 3<i>n</i> + 1 with the common substitute "shortcut" relation (3<i>n</i> + 1)/2, it can be viewed as the restriction to the integers of the smooth real and complex map
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<dl><dd><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(z)={\frac {1}{2}}z\cos ^{2}\left({\frac {\pi }{2}}z\right)+{\frac {1}{2}}(3z+1)\sin ^{2}\left({\frac {\pi }{2}}z\right)}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mi>z</mi><msup><mi>cos</mi><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msup><mo>⁡<!-- ⁡ --></mo><mrow><mo>(</mo><mrow><mrow class="MJX-TeXAtom-ORD"><mfrac><mi>π<!-- π --></mi><mn>2</mn></mfrac></mrow><mi>z</mi></mrow><mo>)</mo></mrow><mo>+</mo><mrow class="MJX-TeXAtom-ORD"><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><mo stretchy="false">(</mo><mn>3</mn><mi>z</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><msup><mi>sin</mi><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msup><mo>⁡<!-- ⁡ --></mo><mrow><mo>(</mo><mrow><mrow class="MJX-TeXAtom-ORD"><mfrac><mi>π<!-- π --></mi><mn>2</mn></mfrac></mrow><mi>z</mi></mrow><mo>)</mo></mrow></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle f(z)={\frac {1}{2}}z\cos ^{2}\left({\frac {\pi }{2}}z\right)+{\frac {1}{2}}(3z+1)\sin ^{2}\left({\frac {\pi }{2}}z\right)}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e449a534b2870da01337c41be5f3643750b0a9d0" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.838ex; width:44.176ex; height:5.176ex;" alt="{\displaystyle f(z)={\frac {1}{2}}z\cos ^{2}\left({\frac {\pi }{2}}z\right)+{\frac {1}{2}}(3z+1)\sin ^{2}\left({\frac {\pi }{2}}z\right)}"></span>,</dd></dl><p>which simplifies to <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 {\frac {1}{4}}(1+4z-(1+2z)\cos(\pi z))}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow class="MJX-TeXAtom-ORD"><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mn>4</mn><mi>z</mi><mo>−<!-- − --></mo><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><mn>2</mn><mi>z</mi><mo stretchy="false">)</mo><mi>cos</mi><mo>⁡<!-- ⁡ --></mo><mo stretchy="false">(</mo><mi>π<!-- π --></mi><mi>z</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle {\frac {1}{4}}(1+4z-(1+2z)\cos(\pi z))}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2dd9526407b6dcd889fb36ad4efd845b9e306f44" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.838ex; width:28.692ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{4}}(1+4z-(1+2z)\cos(\pi z))}"></span>.
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Iterating the above optimized map in the complex plane produces the Collatz fractal.
<a href="//commons.wikimedia.org/w/index.php?title=User:Pokipsy76&amp;action=edit&amp;redlink=1" class="new" title="User:Pokipsy76 (page does not exist)">Pokipsy76</a>
2005-10-21
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