Mathmas Advent Calendar

Day 1

Christmas Wreath

The wreath is formed from two polar equations - the outer and inner edges. Absolute value functions produce the sharp points along the diameter.

$$1/2 - |\cos(6\theta + \pi)|/20 < r < 1 + |\cos(6\theta)|/10$$ for $0 < \theta < 2\pi$

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Day 2

Christmas Tree

$$r = \lfloor 6.05\exp(-3.5(\theta+\pi/2)^2) \rfloor$$ for $-\pi < \theta < 0$

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Day 3

Snowflake

$$r = 1 + 2\cos(6\theta)(2 + \sin^2(24\theta))$$ for $0 < \theta < 2\pi$.

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Day 4

Ornament

$$x = \sin(t)/4 - 2/3 \sin(t)(1+\cos(t)/3)(1 + |\sin(3t + \pi/2)|)$$ $$y = 2\cos(t)$$ for $0 < t < 2\pi$.

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Day 5

Candy Cane

$$\begin{align*} x &= \lfloor t \rfloor \left(\frac{1}{4} - \frac{1}{4}\cos(\pi(t-1)\right) \\ y &= t + \lfloor t \rfloor \left(1- t + \frac{1}{4}\sin(\pi(t-1)\right) \\ \end{align*}$$ for $0 \le t \le 2$

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Day 6

Snowflake Cookie Cutter

This polar equation makes a snowflake shape like a cookie cutter.

$$r = 3 + \cos(6\theta) - \frac{1}{4}\cos(12\theta) + \frac{1}{10}\cos(24\theta)$$ for $0 < \theta < 2\pi$

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Day 7

Heart

$r = 2 - 2\sin(\theta) + \sin(\theta)\sqrt{|\cos(\theta)|} / (\sin(\theta) + \pi/2)$

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Day 8

Star

A clever polar equation for polygons. See reference link for paper describing the equation.

$$r = \frac{\cos{\left(7 \pi/20 \right)}}{\cos{\left(\frac{\arcsin{\left(\sin{\left(5 \theta \right)} \right)}}{5} + \pi/4 \right)}}$$ for $0 < \theta < 2\pi$

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Day 9

Bauble Ornament

The min function results in the union of two implicit equations - a circle and a rectangle.

$$\begin{align*} 0 = \min(&x^2+y^2-1, \\ &|x+y-1|+|x-y+1|- 1/2) \\ \end{align*}$$

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Day 10

Bell

$$\begin{align*} \min(&\max(1-y, \min\left(2x^4+y-1, x^2+y-0.4\right)), \\ & 10 \left(x+0.3\right)^2+10y^2) = 1 \end{align*} $$

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Day 11

Peppermint

The equation defines a set of Archimedean spirals. Parameter t sweeps a spiral from center to outside diameter. Parameter k repeats the spiral through a range of starting angles, while the ceiling function turns it off in the white areas.

$$r = t \lceil \sin(6k) \rceil$$ $$\theta = t+k$$ for $0 < t < 1$ and $0 < k < 2\pi$.

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Day 12

Stocking

$$ \begin{align*} 0 = \min(&x^6 + (2y)^6 - 2, \\ &3x^4 + \frac{1}{4}(y+1.5)^4 - 2, \\ &|x-\frac{1}{2}|^3 + |2(y+5/2)|^2 - 2) \end{align*}$$

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Day 13

Mitten

$$ \begin{align*} x &= 61 t \Phi(0) \\ &+ 61 \Phi(1) \\ &+ \left(17 t^{3} - 231 t^{2} + 831 t - 813\right) \Phi(2) \\ &+ \left(327 t^{3} - 3408 t^{2} + 11697 t - 13188\right) \Phi(3) \\ y &= 28 \left(t - 1\right) \Phi(0) \\ &- \left(38 t^{3} - 204 t^{2} + 255 t - 26\right) \Phi(1) \\ &- \left(43 t^{3} - 162 t^{2} - 414 t + 1468\right) \Phi(2) \\ &- \left(28 t - 140\right) \Phi(3) \\ \Phi(k) &= \max\left(0, \min\left(1, \left\lceil{0.5 - \left|{t - k + 0.5}\right|}\right\rceil\right)\right) \\ \end{align*} $$ for $0 < t < 5$

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Day 14

Snowman

$$\begin{align*} 1 = \min( & x^2 + y^2, \\ & \tfrac{1}{2}x^2 + \tfrac{1}{2}(y+1.8)^2, \\ & \tfrac{1}{4}x^2 + \tfrac{1}{4}(y+4.25)^2, \\ & |\tfrac{1}{2}x^2 + 2y-2| + |\tfrac{1}{2}x^2-2y+2|, \\ & |\tfrac{x}{1.25}+y-1.5| + |\tfrac{x}{1.25}-y+1\tfrac{1}{2}|) \end{align*}$$

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Day 15

Christmas Present

$$|x+y| + |x-y| - 8 = 0$$ $$(y-4)\sin(y-4)/2 - x^2\cos(x/2)/4 = 0$$

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Day 16

Ornament

$$x = \tfrac{1}{4}\sin(t) - \tfrac{1}{2} \sin(t+\pi) (\tfrac{1}{4} + |\sin(3t)|) (\tfrac{1}{2} + \cos(3t))$$ $$y = 2\cos(t)$$ for $0 < t < 2\pi$

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Day 17

Snowflake

One branch of the showflake is formed by the union of three elliptoids. The x1 and y1 equations repeats this branch with rotational symmetry.

$$x_1 = \sqrt{x^2+y^2} \cos(\mod[ \tan^{-1}(y/x), \pi/3] + \pi/3)$$ $$y_1 = \sqrt{x^2+y^2} \sin(\mod[ \tan^{-1}(y/x), \pi/3] + \pi/3)$$ $$\begin{align*} \min(&4000x_1^4 + y_1^4/10 - 1, \\ &30x_1^4 + 6000(y_1-0.9)^4 - 1, \\ &120x_1^4 + 12000(y_1-1.5)^4 - 1) = 0 \end{align*}$$

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Day 18

Gingerbread Man

$$\begin{align*} \min( \\ &4 x^{2} + 4 \left(y - 1\right)^{2} - 1, \\ &\frac{15 \left(x \sqrt{2 - \sqrt{2}} - y \sqrt{\sqrt{2} + 2} + \tfrac{1}{5}\right)^{4}}{2} + \frac{\left(x \sqrt{\sqrt{2} + 2} + y \sqrt{2 - \sqrt{2}} - 1\right)^{4}}{8} - 1 \\ &\frac{15 \left(x \sqrt{2 - \sqrt{2}} + y \sqrt{\sqrt{2} + 2} - \tfrac{1}{5}\right)^{4}}{2} + \frac{\left(x \sqrt{\sqrt{2} + 2} - y \sqrt{2 - \sqrt{2}} + 1\right)^{4}}{8} - 1 \\ &\frac{5 \left(x \sqrt{\sqrt{2} + 2} + y \sqrt{2 - \sqrt{2}}\right)^{4}}{2} + \frac{ \left(- x \sqrt{2 - \sqrt{2}} + y \sqrt{\sqrt{2} + 2} + 1\right)^{4}}{40} - 1 \\ &\frac{5 \left(x \sqrt{\sqrt{2} + 2} - y \sqrt{2 - \sqrt{2}}\right)^{4}}{2} + \frac{\left(x \sqrt{2 - \sqrt{2}} + y \sqrt{\sqrt{2} + 2} + 1\right)^{4}}{40} - 1 \\ &) = 0 \end{align*} $$

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Day 19

Santa Hat

$$\begin{align*} x &= - t^{2} \left(18 t - 27\right) \Phi(0) \\ &- \left(91 t^{3} - 453 t^{2} + 636 t - 283\right) \Phi(1) \\ &+ \left(148 t^{3} - 1158 t^{2} + 2970 t - 2397\right) \Phi(2) \\ &+ \left(103 t^{3} - 1038 t^{2} + 3447 t - 3693\right) \Phi(3) \\ &- \left(7 t^{3} - 99 t^{2} + 456 t - 767\right) \Phi(4) \\ &+ \left(8 t^{3} - 177 t^{2} + 1179 t - 2383\right) \Phi(5) \\ &+ \left(t - 7\right)^{2} \left(13 t - 31\right) \Phi(6) \\ y &= t \left(17 t^{2} - 51 t + 51\right) \Phi(0) \\ &- \left(157 t^{3} - 537 t^{2} + 501 t - 138\right) \Phi(1) \\ &+ \left(83 t^{3} - 543 t^{2} + 1134 t - 732\right) \Phi(2) \\ &- \left(154 t^{3} - 1533 t^{2} + 5040 t - 5505\right) \Phi(3) \\ &- \left(22 t^{3} - 270 t^{2} + 1104 t - 1521\right) \Phi(4) \\ &- \left(33 t^{3} - 573 t^{2} + 3309 t - 6346\right) \Phi(5) \\ &+ \left(38 t^{3} - 717 t^{2} + 4503 t - 9422\right) \Phi(6) \\ \Phi(k) &= \max\left(0, \min\left(1, \left\lceil{0.5 - \left|{t - k + 0.5}\right|}\right\rceil\right)\right) \end{align*}$$ for $0 < t < 8$.

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Day 20

Candle

$$\begin{align*} x& = - t \left(2 t^{2} - 18 t + 24\right) \Phi(0) \\ &- \left(2 t + 6\right) \Phi(1) \\ &- 10 \Phi(2) \\ &- \left(17 t - 41\right) \Phi(3) \\ &+ \left(6 t^{3} - 90 t^{2} + 432 t - 699\right) \Phi(4) \\ &- \left(26 t^{3} - 438 t^{2} + 2448 t - 4501\right) \Phi(5) \\ &+ \left(66 t - 431\right) \Phi(6) \\ &- \left(4 t^{3} - 90 t^{2} + 672 t - 1697\right) \Phi(7) \\ &+ \left(4 t^{3} - 102 t^{2} + 864 t - 2399\right) \Phi(8) \\ &- \left(23 t - 238\right) \Phi(9) \\ &+ 8 \Phi(10) \\ &- \left(2 t - 30\right) \Phi(11) \\ &+ \left(18 t^{3} - 684 t^{2} + 8652 t - 36426\right) \Phi(12) \\ y & = - t \left(- 14 t^{2} + 21 t + 24\right) \Phi(0) \\ &- 31 \Phi(1) \\ &- \left(56 t - 81\right) \Phi(2) \\ &- 87 \Phi(3) \\ &- \left(22 t^{3} - 288 t^{2} + 1248 t - 1705\right) \Phi(4) \\ &+ \left(10 t^{3} - 156 t^{2} + 792 t - 1395\right) \Phi(5) \\ &- 99 \Phi(6) \\ &- \left(6 t^{3} - 138 t^{2} + 1050 t - 2547\right) \Phi(7) \\ &- \left(6 t^{3} - 150 t^{2} + 1242 t - 3315\right) \Phi(8) \\ &- 87 \Phi(9) \\ &+ \left(56 t - 647\right) \Phi(10) \\ &- 31 \Phi(11) \\ &+ \left(22 t^{3} - 825 t^{2} + 10338 t - 43303\right) \Phi(12) \\ \Phi(k) &= \max\left(0, \min\left(1, \left\lceil{0.5 - \left|{t - k + 0.5}\right|}\right\rceil\right)\right) \\ \end{align*}$$ for $0 < t < 14$.

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Day 21

Reindeer

$$\begin{align*} x &= t \left(16 t^{2} - 21 t + 6\right) \Phi(0) \\ &+ \left(1 t^{3} - 27 t^{2} + 108 t - 81\right) \Phi(1) \\ &- \left(3 t^{3} - 15 t^{2} + 18 t - 35\right) \Phi(2) \\ &+ \left(27 t^{3} - 267 t^{2} + 879 t - 928\right) \Phi(3) \\ &- \left(14 t^{3} - 180 t^{2} + 759 t - 1096\right) \Phi(4) \\ &- \left(9 t^{3} - 123 t^{2} + 564 t - 921\right) \Phi(5) \\ &+ \left(61 t^{3} - 1179 t^{2} + 7539 t - 15945\right) \Phi(6) \\ &- \left(30 t^{3} - 669 t^{2} + 4956 t - 12181\right) \Phi(7) \\ &+ \left(87 t^{3} - 2175 t^{2} + 18087 t - 50051\right) \Phi(8) \\ &+ \left(20 t^{3} - 552 t^{2} + 5076 t - 15572\right) \Phi(9) \\ &- \left(24 t^{3} - 807 t^{2} + 8994 t - 33228\right) \Phi(10) \\ &- \left(6 t^{3} - 237 t^{2} + 3066 t - 13032\right) \Phi(11) \\ &- \left(10 t^{3} - 366 t^{2} + 4455 t - 18036\right) \Phi(12) \\ &- \left(38 t^{3} - 1491 t^{2} + 19473 t - 84661\right) \Phi(13) \\ &+ \left(12 t^{3} - 549 t^{2} + 8346 t - 42165\right) \Phi(14) \\ y &= - t \left(- 16 t^{2} + 15 t + 21\right) \Phi(0) \\ &+ \left(34 t^{3} - 141 t^{2} + 189 t - 102\right) \Phi(1) \\ &- \left(6 t^{3} - 48 t^{2} + 138 t - 116\right) \Phi(2) \\ &- \left(11 t^{3} - 123 t^{2} + 471 t - 575\right) \Phi(3) \\ &+ \left(7 t^{3} - 87 t^{2} + 339 t - 457\right) \Phi(4) \\ &- \left(23 t^{3} - 393 t^{2} + 2211 t - 4043\right) \Phi(5) \\ &- \left(38 t^{3} - 738 t^{2} + 4767 t - 10199\right) \Phi(6) \\ &+ \left(14 t^{3} - 321 t^{2} + 2436 t - 6167\right) \Phi(7) \\ &- \left(43 t^{3} - 1107 t^{2} + 9465 t - 26833\right) \Phi(8) \\ &+ \left(5 t^{3} - 114 t^{2} + 837 t - 1976\right) \Phi(9) \\ &+ \left(2 t^{3} - 75 t^{2} + 927 t - 3776\right) \Phi(10) \\ &+ \left(10 t^{3} - 372 t^{2} + 4593 t - 18813\right) \Phi(11) \\ &- \left(1 t^{3} + 18 t^{2} - 921 t + 6717\right) \Phi(12) \\ &- \left(30 t^{3} - 1167 t^{2} + 15111 t - 65147\right) \Phi(13) \\ &+ \left(16 t^{3} - 693 t^{2} + 9996 t - 48015\right) \Phi(14) \\ \Phi(k) &= \max\left(0, \min\left(1, \left\lceil{0.5 - \left|{t - k + 0.5}\right|}\right\rceil\right)\right) \\ \end{align*}$$ for $0 < t < 16$

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Day 22

Snowflake

$$r = k\left( 2 + \frac{\cos(6\theta)}{1 + \sin^2(24\theta)} \right)$$ for $0 < \theta < 2\pi$ and $0 < k < 4$.

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Day 23

Holly

$$x = \tfrac{1}{2} \cos(t) (3 - |\cos(6t)|)$$ $$y = \sin(t) (3 - \tfrac{3}{4} |\sin(2t)|))$$ for $0 < t < 2\pi$ (main leaf only)

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Day 24

Eight-Point Star

A clever polar equation for polygons from

$$r = \frac{\cos{\left(\frac{5 \pi}{16} \right)}}{\sin{\left(\frac{\arcsin{\left(\cos{\left(8 \theta + \pi \right)} \right)}}{8} + \frac{\pi}{8} \right)}}$$ for $0 < \theta < 2\pi$

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Day 25

Santa

$$\begin{align*} x &= 101/9 \\ & + 7/2 \sin( 38/11 - 2\pi t ) + 255/16 \sin( 1/11 - 4\pi t ) \\ & + 67/3 \sin( 22/5 - 6\pi t ) + 107/15 \sin( -3/2 - 8\pi t ) \\ & + 31/12 \sin( 1/16 - 10\pi t ) + \sin( 34/15 - 12\pi t ) \\ & + 32/9 \sin( 71/16 - 14\pi t ) + 2 \sin( 34/15 - 16\pi t ) \\ & + 7/2 \sin( 31/13 - 18\pi t ) + 1/15 \sin( 41/11 - 20\pi t ) \\ & + 1/2 \sin( 3/16 - 22\pi t ) + 20/9 \sin( 16/9 - 24\pi t ) \\ & + 15/16 \sin( 1/3 - 26\pi t ) + 10/11 \sin( 1/14 - 28\pi t ) \\ & + 9/13 \sin( 37/10 - 30\pi t ) + 4/3 \sin( 19/6 - 32\pi t ) \\ & + 7/9 \sin( 21/13 - 34\pi t ) + 1/7 \sin( 8/9 - 36\pi t ) \\ & + 7/16 \sin( -23/15 - 38\pi t ) + 7/15 \sin( -8/11 - 40\pi t ) \\ & + 5/9 \sin( 3 - 42\pi t ) + 9/16 \sin( 3/8 - 44\pi t ) \\ & + 8/15 \sin( 33/8 - 46\pi t ) + 1/4 \sin( 5/4 - 48\pi t ) \\ & + 1/7 \sin( -13/14 - 50\pi t ) + 1/9 \sin( 4/3 - 52\pi t ) \\ & + 4/13 \sin( 8/9 - 54\pi t ) + 5/12 \sin( -1/8 - 56\pi t ) \\ & + 1/7 \sin( -23/16 - 58\pi t ) + 5/12 \sin( 25/12 - 60\pi t ) \\ & + 3/13 \sin( 10/13 - 62\pi t ) + 2/15 \sin( 1/16 - 64\pi t ) \\ & + 1/4 \sin( 33/13 - 66\pi t ) + 1/10 \sin( 54/13 - 68\pi t ) \\ & + 1/14 \sin( 7/6 - 70\pi t ) + 1/16 \sin( 5/2 - 72\pi t ) \\ & + 1/4 \sin( 16/9 - 74\pi t ) \\ y &= 21/8 \\ & + 12 \sin( 16/13 - 2\pi t ) + 128/9 \sin( -4/9 - 4\pi t ) \\ & + 125/16 \sin( 24/13 - 6\pi t ) + 133/15 \sin( 3/16 - 8\pi t ) \\ & + 68/13 \sin( -3/13 - 10\pi t ) + 52/15 \sin( -5/9 - 12\pi t ) \\ & + 23/9 \sin( 48/11 - 14\pi t ) + 34/15 \sin( 51/11 - 16\pi t ) \\ & + 1/2 \sin( 21/13 - 18\pi t ) + 7/5 \sin( 25/9 - 20\pi t ) \\ & + 17/14 \sin( 41/10 - 22\pi t ) + 17/12 \sin( 10/11 - 24\pi t ) \\ & + 13/7 \sin( 29/12 - 26\pi t ) + \sin( 19/5 - 28\pi t ) \\ & + 11/12 \sin( 8/5 - 30\pi t ) + 2/3 \sin( 41/15 - 32\pi t ) \\ & + \sin( -4/3 - 34\pi t ) + 7/8 \sin( 35/9 - 36\pi t ) \\ & + 1/2 \sin( 15/13 - 38\pi t ) + 7/16 \sin( 7/15 - 40\pi t ) \\ & + 9/13 \sin( 23/10 - 42\pi t ) + 5/14 \sin( 65/16 - 44\pi t ) \\ & + 7/6 \sin( 29/16 - 46\pi t ) + 4/15 \sin( 11/5 - 48\pi t ) \\ & + 2/3 \sin( -1 - 50\pi t ) + 2/7 \sin( 14/11 - 52\pi t ) \\ & + 11/14 \sin( 3 - 54\pi t ) + 11/16 \sin( 19/7 - 56\pi t ) \\ & + 7/8 \sin( 33/16 - 58\pi t ) + 4/11 \sin( 1/13 - 60\pi t ) \\ & + 1/6 \sin( 63/16 - 62\pi t ) + 5/9 \sin( 36/13 - 64\pi t ) \\ & + 4/15 \sin( 19/11 - 66\pi t ) + 4/15 \sin( 1/2 - 68\pi t ) \\ & + 4/11 \sin( -23/15 - 70\pi t ) + 7/16 \sin( 38/15 - 72\pi t ) \\ & + 13/14 \sin( 3 - 74\pi t ) \end{align*}$$ for $0 < t < 2\pi$

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Merry Mathmas

A mathematical Advent Calendar of Christmas shapes and their equations

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