Called <a href="https://en.wikipedia.org/wiki/Regular_polygon" class="extiw" title="w:Regular polygon">regular</a> dodecagons, a few <a href="https://en.wiktionary.org/wiki/polygon#Noun" class="extiw" title="wiktionary:polygon">polygons</a> have twelve <a href="https://en.wiktionary.org/wiki/side#Noun" class="extiw" title="wiktionary:side">sides</a> within the image.  Consider two regular <a href="https://en.wiktionary.org/wiki/dodecagon#Noun" class="extiw" title="wiktionary:dodecagon">dodecagons</a> sharing <a href="https://en.wiktionary.org/wiki/center#Noun" class="extiw" title="wiktionary:center">their center</a> and all <a href="https://en.wiktionary.org/wiki/vertices#Noun" class="extiw" title="wiktionary:vertices">their vertices.</a>  If they are different, one of them <a href="https://en.wikipedia.org/wiki/Convex_set" class="extiw" title="w:Convex set">is convex</a>, the other <a href="https://en.wikipedia.org/wiki/Regular_polygon#Regular_star_polygons" class="extiw" title="w:Regular polygon">is stellated.</a>  A side of any of them is <a href="https://en.wiktionary.org/wiki/diagonal#Noun" class="extiw" title="wiktionary:diagonal">a diagonal</a> of the other.
<p>The hours are numbered from 1 to 12.  There is an <a href="https://en.wikipedia.org/wiki/Bijection" class="extiw" title="w:Bijection">one-to-one correspondence</a> between the sides of the clock and these integers, written beside the twelve sides.  Twelve hairpin bends can be also numbered from 1 to 12, along an imaginary decagonal race track, the plan of which is on the clock face.  Such an object is called <a href="https://en.wikipedia.org/wiki/Knot_theory" class="extiw" title="w:Knot theory">a knot</a> in mathematics.  We meet no intersection along this loop, but forty-eight bridges.  Every straight line crosses eight other parts, passing alternatively under and above.  We see the same knot on <a href="//commons.wikimedia.org/wiki/File:%22Knot_with_48_crossings%22_on_a_clock_face.svg#file" title='File:"Knot with 48 crossings" on a clock face.svg'>another version.</a>  This knot imparts an information:  a stellated polygon is really <a href="https://en.wikipedia.org/wiki/Polygon" class="extiw" title="w:Polygon">a polygon.</a>  In other words, it is a closed <a href="https://en.wikipedia.org/wiki/Polygonal_chain" class="extiw" title="w:Polygonal chain">polygonal chain:</a>  the twelve sides of the star can be traced without lifting the pencil.
</p><p>For example, if we start at the bend numbered one, we come back to the same bend numbered one after covering the whole track, exactly after five turns around the center.  <a href="https://en.wikipedia.org/wiki/Rotation_(mathematics)" class="extiw" title="w:Rotation (mathematics)">This rotation</a> can be seen as the <a href="https://en.wikipedia.org/wiki/Identity_function" class="extiw" title="w:Identity function">"transformation into the identical one"</a>, in other words:  a rotation of which the angle is zero <a href="https://en.wikipedia.org/wiki/Congruence_relation" class="extiw" title="w:Congruence relation">modulo 360 degrees.</a>  By passing from a bend of the track to the next one, we turn 150 degrees around the center:  <span style="white-space:nowrap">5 × 30° = 150°. </span> <span style="white-space:nowrap">And  12 × 150° = 5 × 360° = 0° </span> modulo 360°.  Provided that their centers coincide, and that their angles are equal modulo 360 degrees, several rotations are identical.
</p><p>By turning clockwise, we pass through the twelve bends:  <span style="white-space:nowrap">1, 6, 11, 4, 9, 2, 7, 12, 5, 10, 3, 8. </span> Whatever the integer from which we start, <a href="https://en.wiktionary.org/wiki/up_to#Preposition" class="extiw" title="wiktionary:up to">up to</a> congruence modulo 12, we pass through the terms of an <a href="https://en.wikipedia.org/wiki/Arithmetic_progression" class="extiw" title="w:Arithmetic progression">arithmetic progression</a> with <a href="https://en.wiktionary.org/wiki/common_difference" class="extiw" title="wiktionary:common difference">common difference</a>  5  or  7,  depending on the sense of rotation, either clockwise or the inverse sense.  The common differences  –5  and  +7  are equivalent:  <span style="white-space:nowrap">–5 = +7  modulo 12.</span>
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If it is now nine o'clock, it will be noon in three hours.  Up to congruence modulo 12, the sequence  (9; 0; 3; 6)  is an arithmetic progression with common difference  +3  or  –9.  The hairpin bends numbered  <span style="white-space:nowrap">9, 0, 3, 6, </span> are the vertices of a square.  Up to <a href="https://en.wikipedia.org/wiki/Similarity_(geometry)" class="extiw" title="w:Similarity (geometry)">similarity</a>, there is only one regular <a href="https://en.wiktionary.org/wiki/quadrilateral#Noun" class="extiw" title="wiktionary:quadrilateral">quadrilateral:</a>  the square, denoted by {4,1} <a href="https://en.wikipedia.org/wiki/Schl%C3%A4fli_symbol" class="extiw" title="w:Schläfli symbol">in Schläfli notation</a>, not {12,3}.  One of the following notations denotes a regular dodecagon:  {12,1} for a convex one, otherwise {12,5}.<br> 
<a href="//commons.wikimedia.org/wiki/User:Baelde" title="User:Baelde">Baelde</a>
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