Fractal Kitty<p>I love Pisano Periods so I played with a draft visualization I'd like to explore more in the future:</p><p><a href="https://codepen.io/fractalkitty/pen/abeYVGK" rel="nofollow noopener" translate="no" target="_blank"><span class="invisible">https://</span><span class="ellipsis">codepen.io/fractalkitty/pen/ab</span><span class="invisible">eYVGK</span></a></p><p><a href="https://mathstodon.xyz/tags/mathober" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>mathober</span></a> <a href="https://mathstodon.xyz/tags/mathober2024" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>mathober2024</span></a> <a href="https://mathstodon.xyz/tags/pisano" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>pisano</span></a> <a href="https://mathstodon.xyz/tags/fibonacci" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>fibonacci</span></a></p>
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#mathober2024
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Fractal Kitty<p>I did not go with the empty set:</p><p><a href="https://codepen.io/fractalkitty/pen/KKOopZP" rel="nofollow noopener" translate="no" target="_blank"><span class="invisible">https://</span><span class="ellipsis">codepen.io/fractalkitty/pen/KK</span><span class="invisible">OopZP</span></a></p><p><a href="https://mathstodon.xyz/tags/mathober" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>mathober</span></a> <a href="https://mathstodon.xyz/tags/mathober2024" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>mathober2024</span></a></p>
Oscar Cunningham<p><a href="https://mathstodon.xyz/tags/Mathober" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Mathober</span></a> <a href="https://mathstodon.xyz/tags/Mathober2024" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Mathober2024</span></a></p><p>The prompt for day 6 was 'Nine-Point Circle'. In geometry, the nine-point circle of a given triangle is a circle that passes through the midpoint of each side, the foot of each altitude, and the points halfway between each vertex and the orthocentre.</p><p>Feuerbach proved that the nine-point circle is tangent to the incircle and each of the excircles. These circles are in turn tangent to each of the triangle's three sides. In fact, each of these objects can move at the same speed while touching the others without slipping.</p><p><a href="https://mathstodon.xyz/tags/Math" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Math</span></a> <a href="https://mathstodon.xyz/tags/Maths" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Maths</span></a> <a href="https://mathstodon.xyz/tags/Mathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Mathematics</span></a> <a href="https://mathstodon.xyz/tags/Geometry" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Geometry</span></a> <a href="https://mathstodon.xyz/tags/Geogebra" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Geogebra</span></a></p>
Fractal Kitty<p>9-pointed circle doodle. <a href="https://mathstodon.xyz/tags/mathober" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>mathober</span></a> <a href="https://mathstodon.xyz/tags/mathober2024" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>mathober2024</span></a> <a href="https://mathstodon.xyz/tags/mtbos" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>mtbos</span></a> <a href="https://mathstodon.xyz/tags/geometry" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>geometry</span></a></p>
Fractal Kitty<p>Ok - I am in love with 9-point circles: </p><p><a href="https://codepen.io/fractalkitty/full/WNVrNNy" rel="nofollow noopener" translate="no" target="_blank"><span class="invisible">https://</span><span class="ellipsis">codepen.io/fractalkitty/full/W</span><span class="invisible">NVrNNy</span></a></p><p><a href="https://mathstodon.xyz/tags/mathober" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>mathober</span></a> <a href="https://mathstodon.xyz/tags/triangle" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>triangle</span></a> <a href="https://mathstodon.xyz/tags/ninePoint" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>ninePoint</span></a> <a href="https://mathstodon.xyz/tags/p5js" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>p5js</span></a> <a href="https://mathstodon.xyz/tags/mathober2024" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>mathober2024</span></a></p>
Oscar Cunningham<p><a href="https://mathstodon.xyz/tags/Mathober" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Mathober</span></a> <a href="https://mathstodon.xyz/tags/Mathober2024" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Mathober2024</span></a></p><p>The prompt for day 5 was 'Integer Partitions'. In number theory, a partition is a way to write an integer as a sum of positive integers. For example, there are 5 partitions of 4, given by 1+1+1+1, 2+1+1, 2+2, 3+1 and 4. The partition function is the name given to the function that counts how many partitions a given integer has. So the above examples show that p(4)=5. The first few values of the partition function are 1, 1, 2, 3, 5, 7, 11, ....</p><p>In 1937, Hans Rademacher found a complicated formula for the partition function in the form of an infinite series. You can see the full formula here <a href="https://en.wikipedia.org/wiki/Partition_function_(number_theory)#Approximation_formulas" rel="nofollow noopener" translate="no" target="_blank"><span class="invisible">https://</span><span class="ellipsis">en.wikipedia.org/wiki/Partitio</span><span class="invisible">n_function_(number_theory)#Approximation_formulas</span></a>. One interesting feature of this formula is that it allows you to calculate a value of p(x) even when x is not an integer. This was explored and graphed a bit by Fredrik Johansson over at <a href="https://mathoverflow.net/questions/366733/does-rademachers-convergent-series-for-pn-define-an-analytic-function/366805#366805" rel="nofollow noopener" translate="no" target="_blank"><span class="invisible">https://</span><span class="ellipsis">mathoverflow.net/questions/366</span><span class="invisible">733/does-rademachers-convergent-series-for-pn-define-an-analytic-function/366805#366805</span></a>. He points out that when x is n+1/2 for natural n, the infinite series is zero for every term except the first. This then gives you a closed form expression, which he doesn't actually write out because it's awful:</p><p>p(x) = (√(2/3)cosh(π√(2/3)√(x-1/24)) - sinh(π√(2/3)√(x-1/24))/(π√(x-1/24)))/(2√2(x-1/24))</p><p>It's interesting that p(n+1/2) has this closed form formula, because no such formula is known for p(n) itself.</p><p>Of course it would be quite irresponsible to say 'There are 0.8458... ways to write 1/2 as the sum of natural numbers', so I won't.</p><p><a href="https://mathstodon.xyz/tags/Math" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Math</span></a> <a href="https://mathstodon.xyz/tags/Maths" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Maths</span></a> <a href="https://mathstodon.xyz/tags/Mathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Mathematics</span></a> <a href="https://mathstodon.xyz/tags/NumberTheory" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>NumberTheory</span></a> <a href="https://mathstodon.xyz/tags/Combinatorics" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Combinatorics</span></a></p>
Oscar Cunningham<p><a href="https://mathstodon.xyz/tags/Mathober" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Mathober</span></a> <a href="https://mathstodon.xyz/tags/Mathober2024" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Mathober2024</span></a></p><p>The prompt for day 4 was 'Form'. In algebra, a multilinear map is a function from several vector spaces to another vector space which is linear in each argument. Just as a linear map can be represented by a grid of numbers in a matrix, a multilinear map can be represented by a multidimensional grid of numbers: a tensor. A 'form' is the function you get by taking a multilinear map from V×...×V to the scalars and evaluating it with each of its arguments the same. In this way you get a *non*linear map from V to the scalars.</p><p>The image shows a degree 50 form on ℝ³ evaluated on the unit sphere. The form was randomly chosen with each of the 3⁵⁰ entries in its tensor having a standard normal distribution. </p><p><a href="https://mathstodon.xyz/tags/Math" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Math</span></a> <a href="https://mathstodon.xyz/tags/Maths" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Maths</span></a> <a href="https://mathstodon.xyz/tags/Mathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Mathematics</span></a> <a href="https://mathstodon.xyz/tags/Algebra" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Algebra</span></a></p>
illestpreacha<p>SankeyPartition</p><p>Blogpost for More Images/Code : <a href="https://blog.illestpreacha.com/mathober2024partition" rel="nofollow noopener" translate="no" target="_blank"><span class="invisible">https://</span><span class="ellipsis">blog.illestpreacha.com/mathobe</span><span class="invisible">r2024partition</span></a></p><p>Mathober made by <span class="h-card" translate="no"><a href="https://mathstodon.xyz/@fractalkitty" class="u-url mention" rel="nofollow noopener" target="_blank">@<span>fractalkitty</span></a></span> <br><a href="https://post.lurk.org/tags/mathober" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>mathober</span></a> <a href="https://post.lurk.org/tags/mathart" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>mathart</span></a> <a href="https://post.lurk.org/tags/mathober2024" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>mathober2024</span></a> <a href="https://post.lurk.org/tags/mathober5" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>mathober5</span></a> <a href="https://post.lurk.org/tags/mathober30" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>mathober30</span></a> <a href="https://post.lurk.org/tags/commutative" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>commutative</span></a> <a href="https://post.lurk.org/tags/partition" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>partition</span></a></p><p>For my fifth sketch of Mathober, Which is coded in <a href="https://post.lurk.org/tags/MermaidJs" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>MermaidJs</span></a>, SankeyPartition will be using the 5th prompt of Mathober 2024: Integer Partition and 30th prompt: Commutative. This is done through the Sankey Diagram, where each curve is representing a different partition that is commutative and additional glitching is created with <a href="https://post.lurk.org/tags/Glitchlab" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Glitchlab</span></a>.</p><p><a href="https://post.lurk.org/tags/Poetry" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Poetry</span></a></p><p>Partitioning the Connections<br>Connecting the Partitions<br>Commutative to the <br>As these numbers are a community </p><p><a href="https://post.lurk.org/tags/creativecoding" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>creativecoding</span></a> <a href="https://post.lurk.org/tags/coding" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>coding</span></a> <a href="https://post.lurk.org/tags/dataart" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>dataart</span></a><br><a href="https://post.lurk.org/tags/newmedia" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>newmedia</span></a> <a href="https://post.lurk.org/tags/diagram" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>diagram</span></a> <a href="https://post.lurk.org/tags/animation" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>animation</span></a></p>
Oscar Cunningham<p><a href="https://mathstodon.xyz/tags/Mathober" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Mathober</span></a> <a href="https://mathstodon.xyz/tags/Mathober2024" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Mathober2024</span></a></p><p>The prompt for day 3 was 'Potential'. The animation shows the potential of a field produced by some particles that repel each other. The particles accelerate according to the gradient of this potential.</p><p><a href="https://mathstodon.xyz/tags/Math" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Math</span></a> <a href="https://mathstodon.xyz/tags/Maths" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Maths</span></a> <a href="https://mathstodon.xyz/tags/Mathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Mathematics</span></a> <a href="https://mathstodon.xyz/tags/Physics" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Physics</span></a></p>
Oscar Cunningham<p><a href="https://mathstodon.xyz/tags/Mathober" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Mathober</span></a> <a href="https://mathstodon.xyz/tags/Mathober2024" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Mathober2024</span></a></p><p>The prompt for day 2 was 'Inverted'. In geometry, an inversion in a circle is a transformation that swaps the inside and outside of the circle. Every point on the plane is moved towards or away from the centre of the circle so that its new distance from the centre is inversely proportional to its old distance.</p><p>We can use this transformation to play a 'chaos game'. Start at any point in the plane and do one of three things at random. Either move halfway towards one of the two red points, or invert your position in the circle. After repeating this for 1000000 steps the trace of your positions forms a beautiful fractal.</p><p><a href="https://mathstodon.xyz/tags/Math" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Math</span></a> <a href="https://mathstodon.xyz/tags/Maths" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Maths</span></a> <a href="https://mathstodon.xyz/tags/Mathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Mathematics</span></a> <a href="https://mathstodon.xyz/tags/Fractal" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Fractal</span></a> <a href="https://mathstodon.xyz/tags/Geometry" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Geometry</span></a> <a href="https://mathstodon.xyz/tags/EuclideanGeometry" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>EuclideanGeometry</span></a></p>
Oscar Cunningham<p><a href="https://mathstodon.xyz/tags/Mathober" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Mathober</span></a> <a href="https://mathstodon.xyz/tags/Mathober2024" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Mathober2024</span></a></p><p>The prompt for day 1 was 'Tangent'. So I decided to draw an Apollonian gasket.</p><p><a href="https://mathstodon.xyz/tags/Math" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Math</span></a> <a href="https://mathstodon.xyz/tags/Maths" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Maths</span></a> <a href="https://mathstodon.xyz/tags/Mathematics" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>Mathematics</span></a></p>
Fractal Kitty<p>I also doodled a tangent today:<br><a href="https://mathstodon.xyz/tags/mathober2024" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>mathober2024</span></a> <a href="https://mathstodon.xyz/tags/tangent" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>tangent</span></a></p>
Fractal Kitty<p>It's <a href="https://mathstodon.xyz/tags/mathober2024" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>mathober2024</span></a>!!! </p><p>Here is the website for prompts: <br><a href="https://mathober.com/" rel="nofollow noopener" translate="no" target="_blank"><span class="invisible">https://</span><span class="">mathober.com/</span><span class="invisible"></span></a></p><p>Here is my first sketch:<br><a href="https://openprocessing.org/sketch/2376074" rel="nofollow noopener" translate="no" target="_blank"><span class="invisible">https://</span><span class="ellipsis">openprocessing.org/sketch/2376</span><span class="invisible">074</span></a></p><p>or on codepen:<br><a href="https://codepen.io/fractalkitty/pen/oNKjZqJ/e30039a4d7b8931096da090f87a8231b" rel="nofollow noopener" translate="no" target="_blank"><span class="invisible">https://</span><span class="ellipsis">codepen.io/fractalkitty/pen/oN</span><span class="invisible">KjZqJ/e30039a4d7b8931096da090f87a8231b</span></a></p><p>I can't wait to see what everyone does this year!</p><p><a href="https://mathstodon.xyz/tags/mathober" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>mathober</span></a> <a href="https://mathstodon.xyz/tags/tangent" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>tangent</span></a> <a href="https://mathstodon.xyz/tags/mathart" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>mathart</span></a> <a href="https://mathstodon.xyz/tags/mtbos" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>mtbos</span></a></p>
Fractal Kitty<p>This year's prompts!</p><p><a href="https://mathstodon.xyz/tags/mathober2024" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>mathober2024</span></a> <a href="https://mathstodon.xyz/tags/mathober" class="mention hashtag" rel="nofollow noopener" target="_blank">#<span>mathober</span></a></p><p><a href="https://fractalkitty.com/2024/09/10/mathober-2024/" rel="nofollow noopener" translate="no" target="_blank"><span class="invisible">https://</span><span class="ellipsis">fractalkitty.com/2024/09/10/ma</span><span class="invisible">thober-2024/</span></a></p>
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