<a href="https://en.wikipedia.org/wiki/Blichfeldts_theorem" class="extiw" title="en:Blichfeldts theorem">Blichfeldts theorem</a>, in the form that every plane set of area <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 A}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi>A</mi></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle A}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="A"></span> (here, an ellipse with area <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 \pi }"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi>π<!-- π --></mi></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle \pi }</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="\pi"></span>) contains at least <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 \lceil A\rceil }"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mo fence="false" stretchy="false">⌈<!-- ⌈ --></mo><mi>A</mi><mo fence="false" stretchy="false">⌉<!-- ⌉ --></mo></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle \lceil A\rceil }</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2dc1380e50dd2ccb6589b858502003720f0f2963" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:3.808ex; height:2.843ex;" alt="{\displaystyle \lceil A\rceil }"></span> points (here, <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 \lceil \pi \rceil =4}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mo fence="false" stretchy="false">⌈<!-- ⌈ --></mo><mi>π<!-- π --></mi><mo fence="false" stretchy="false">⌉<!-- ⌉ --></mo><mo>=</mo><mn>4</mn></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle \lceil \pi \rceil =4}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b44610e861a8d939e7d4c5718fb85699d2ac3f1" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:7.658ex; height:2.843ex;" alt="{\displaystyle \lceil \pi \rceil =4}"></span>  points), all of whose coordinates differ from each other by integers. The theorem is proved by cutting the set up by squares of the integer grid (top), translating each piece by an integer translation vector into a single unit square, finding a point in that unit square that is covered by many pieces (middle), and using the preimages of this point as the desired points (bottom).
<a href="//commons.wikimedia.org/wiki/User:David_Eppstein" class="mw-redirect" title="User:David Eppstein">David Eppstein</a>
<span class="int-own-work" lang="en">Own work</span>