{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "## Part 2: Drawing the Heptagon" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The heptagon numbers we defined in [Part 1](https://nbviewer.org/github/vorth/ipython/blob/master/heptagons/HeptagonNumbers.ipynb) are obviously useful for defining lengths of lines in a figure like the one below, since all of those line lengths are related to the $\\rho$ and $\\sigma$ ratios. But will heptagon numbers suffice for the coordinates we must give to a graphics library, to draw line segments on the screen? The answer turns out to be yes, though we'll need one little trick to make it work.\n", "\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Finding the Heptagon Vertices" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "To start with, we will try to construct coordinates for the seven vertices of the heptagon. We will label the points as in the figure below, $P0$ through $P6$.\n", "\n", "\n", "\n", "For convenience, we can say that $P0$ is the origin of our coordinate system, so it will have coordinates $(0,0)$. If the heptagon edge length is one, then the coordinates of point $P1$ are clearly $(1,0)$. But now what? We can use the Pythogorean theorem to find the coordinates of point $P4$, but we end up with $(1/2,a)$, where\n", "\n", "$$a = \\sqrt{\\sigma^2 - \\frac{1}{4}}\n", " = \\sqrt{\\frac{3}{4} + \\rho + \\sigma} $$\n", " \n", "This is annoying, since we have no way to take the square root of a heptagon number! Fortunately, there is an easier way.\n", "\n", "Suppose we abandon our *usual* Cartesian coordinate frame, and use one that works a little more naturally for us? We can use a different frame of reference, one where the \"y\" axis is defined as the line passing through points $P0$ and $P4$. We can then model all the heptagon vertices quite naturally, and adjust for the modified frame of reference when we get to the point of drawing.\n", "\n", "For the sake of further convenience, let us scale everything up by a factor of $\\sigma$. This makes it quite easy to write down the coordinates of all the points marked on the diagram above. Notice the three points I have included in the interior of the heptagon. Those points divide the horizontal and vertical diagonals into $\\rho:\\sigma:1$ and $\\rho:\\sigma$ proportions. So now we have:\n", "\n", "|point|coordinates|\n", "|-----|-----------|\n", "|$P0$|$(0,0)$|\n", "|$P1$|$(\\sigma,0)$|\n", "|$P2$|$(1+\\sigma,\\rho)$|\n", "|$P3$|$(\\sigma,\\rho+\\sigma)$|\n", "|$P4$|$(0,1+\\rho+\\sigma)$|\n", "|$P5$|$(-\\rho,\\rho+\\sigma)$|\n", "|$P6$|$(-\\rho,\\rho)$|\n", "\n", "If we render our heptagon and heptagrams with these coordinates, *ignoring* the fact that we used an unusual coordinate frame, we get a skewed heptagon:\n", "\n", "\n", "\n", "This figure is certainly not *regular* in any usual sense, since\n", "the edge lengths and angles vary, but we could refer to it as an **affine regular heptagon** (and associated heptagrams). In linear algebra, an *affine* transformation is one that preserves parallel lines and ratios along lines. Those are exactly the properties that we took advantage of in capturing our coordinates.\n", "\n", "Although we have not yet discussed how we will accommodate the skew coordinate frame we've used, let's take a look at the Python code to capture the vertices used above. Note that a point is represented simply as a pair of heptagon numbers, using Python's tuple notation.\n" ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "collapsed": false }, "outputs": [], "source": [ "# load the definitions from the previous notebook\n", "%run HeptagonNumbers.py\n", "\n", "# represent points or vertices as pairs of heptagon numbers\n", "\n", "p0 = ( zero, zero )\n", "p1 = ( sigma, zero )\n", "p2 = ( sigma+1, rho )\n", "p3 = ( sigma, rho*sigma )\n", "p4 = ( zero, sigma*sigma )\n", "p5 = ( -rho, rho*sigma )\n", "p6 = ( -rho, rho )\n", "\n", "heptagon = [ p0, p1, p2, p3, p4, p5, p6 ]\n", "\n", "heptagram_rho = [ p0, p2, p4, p6, p1, p3, p5 ]\n", "\n", "heptagram_sigma = [ p0, p3, p6, p2, p5, p1, p4 ]\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Evaluating a Heptagon Number" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "We are almost ready to look at Python code for drawing these figures. The last step is to provide a mapping from heptagon numbers to real numbers, since any drawing library requires us to provide points as $(x,y)$ pairs, where $x$ and $y$ are floating-point numbers. This is easy, of course: we just evaluate the expression $a+b\\rho+c\\sigma$, using predefined values for $\\rho$ and $\\sigma$. But what are those values?\n", "\n", "We can easily derive them from a tiny bit of trigonometry, looking at our heptagon-plus-heptagrams figure again.\n", "\n", "\n", "\n", "I have rotated the heptagon a bit, so we see angles presented in the traditional way, with zero radians corresponding to the X axis, to the right, and considering angles around the point $A$. We can find $\\sigma$ using the right triangle $\\triangle ABC$. If our heptagon has edge length of one, then line segment $AB$ has length $\\sigma$, by definition, and line segment $BC$ has length $1/2$. Remembering the formula for the sine function (opposite over hypotenuse), we can see that\n", "\n", "$$\\sin \\angle CAB = \\frac{1/2}{\\sigma} = \\frac{1}{2\\sigma}$$\n", "\n", "which means that\n", "\n", "$$\\sigma = \\frac{1}{2\\sin \\angle CAB}$$\n", "\n", "Now we just need to know what angle $\\angle CAB$ is. Here, you can have some fun by convincing yourself that $\\angle CAB$ is equal to $\\pi/14$ radians. As a hint, first use similar triangles to show that all those narrow angles at the heptagon vertices are equal to $\\pi/7$. In any case, we have our value for $\\sigma$:\n", "\n", "$$\\sigma = \\frac{1}{2\\sin{\\frac{\\pi}{14}}} $$\n", "\n", "Computing $\\rho$ is just as easy. It is convenient to use triangle $\\triangle ADE$, with the following results:\n", "\n", "$$\\frac{\\rho}{2} = \\sin{\\angle EAD} = \\sin{\\frac{5\\pi}{14}}$$\n", "\n", "$$\\rho = 2\\sin{\\frac{5\\pi}{14}}$$\n", "\n", "These values for $\\rho$ and $\\sigma$ were already captured as constants in [Part 1](http://nbviewer.ipython.org/github/vorth/ipython/blob/master/heptagons/HeptagonNumbers.ipynb). Here we will just print out the values, and define a rendering function that produces a vector of floats from a vector of heptagon numbers.\n" ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "sigma = 2.24697960372\n", "rho = 1.8019377358\n" ] } ], "source": [ "print \"sigma = \" + str( HeptagonNumber.sigma_real )\n", "\n", "print \"rho = \" + str( HeptagonNumber.rho_real )\n", "\n", "# This function maps from a pair of heptagon numbers to\n", "# a pair of floating point numbers (approximating real numbers)\n", "def render( v ):\n", " x, y = v\n", " return [ float(x), float(y) ]\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Python Code for Drawing" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Although we have not yet discussed how we will accommodate the skew coordinate frame we've used, let's take a look at the Python code to render the skewed heptagon shown earlier. I won't go over the details of the `matplotlib` drawing library, since you can explore the online documentation. The thing to note is the `drawPolygon` function, which takes an array of heptagon number pairs as vertices, then renders them and draws a path connecting them." ] }, { "cell_type": "code", "execution_count": 15, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Populating the interactive namespace from numpy and matplotlib\n" ] }, { "data": { "image/png": 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AHaAf8BvwYgWPWe15NPGgxr01iN58kW+/9SY3V6F//0QKChTc3S157TU3Ll2q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\n4cYNWScuKwcHC4YONa1ijr/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44IMaBnmb3hxD2DUsjLSFC1G0FdttL8rGEG9BZ//zvzaA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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "%pylab inline\n", "\n", "import matplotlib.pyplot as plt\n", "import matplotlib.path as mpath\n", "import matplotlib.patches as mpatches\n", "Path = mpath.Path\n", "\n", "def drawPolygon( polygonVerts, color, mapping=render ):\n", " n = len( polygonVerts )\n", " codes = [ Path.MOVETO ]\n", " verts = []\n", " verts .append( mapping( polygonVerts[ 0 ] ) )\n", " for i in range(1,n+1):\n", " codes.append ( Path.LINETO )\n", " verts.append ( mapping( polygonVerts[ i % n ] ) )\n", " path = mpath.Path( verts, codes )\n", " return mpatches.PathPatch( path, facecolor='none', edgecolor=color )\n", "\n", "def drawHeptagrams( mapping=render ):\n", " fig = plt.figure(figsize=(8,8))\n", " ax = fig.add_subplot(111)\n", " ax.add_patch( drawPolygon( heptagon,'#dd0000', mapping ) )\n", " ax.add_patch( drawPolygon( heptagram_rho,'#990099', mapping ) )\n", " ax.add_patch( drawPolygon( heptagram_sigma,'#0000dd', mapping ) )\n", " ax.set_xlim(-3,4)\n", " ax.set_ylim(-1,6)\n", "\n", "drawHeptagrams()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Correcting the Skew" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now all that remains is to straighten up our heptagon by applying a shear transformation. This is simpler than it sounds, using the most basic technique of linear algebra, a change of basis.\n", "\n", "The trick is to represent points as a column vectors:\n", "\n", "$$(x,y) \\Rightarrow \\begin{bmatrix}x \\\\ y\\end{bmatrix}$$\n", "\n", "Now, consider the vectors corresponding to the natural basis vectors we used to construct the heptagon vertices:\n", "\n", "$$ \\begin{bmatrix}1 \\\\ 0\\end{bmatrix}, \\begin{bmatrix}0 \\\\ 1\\end{bmatrix} $$\n", "\n", "What should our transformation do to these two vectors? We don't need any change in the X-axis direction, so we'll leave that one alone.\n", "\n", "$$ \\begin{bmatrix}1 \\\\ 0\\end{bmatrix} \\Rightarrow \\begin{bmatrix}1 \\\\ 0\\end{bmatrix} $$\n", "\n", "For the Y-axis, we must determine what point in a traditional cartesian plane corresponds to our \"vertical\" basis vector. We can find that by applying a bit of trigonometry, as we did when deriving $\\rho$ and $\\sigma$ earlier. The result is:\n", "\n", "$$ \\begin{bmatrix}1 \\\\ 0\\end{bmatrix} \\Rightarrow \\begin{bmatrix} \\frac{1}{2\\sigma} \\\\ \\sin\\frac{3}{7}\\pi\\end{bmatrix} $$\n", "\n", "It turns out that specifying those two transformations is equivalent to specifying the transformation of any initial vector. We use our transformed basis vectors as the column of a matrix, and write:\n", "\n", "$$\\begin{bmatrix}x' \\\\ y'\\end{bmatrix} = \\begin{bmatrix}1 & \\frac{1}{2\\sigma} \\\\0 & \\sin\\frac{3}{7}\\pi\\end{bmatrix}\n", "\\begin{bmatrix}x \\\\ y\\end{bmatrix}$$\n", "\n", "Again, this is simpler than it looks. It is just a way of writing one equation that represents two equations:\n", "\n", "$$ x' = x + y \\frac{1}{2\\sigma} $$\n", "\n", "$$ y' = y \\sin\\frac{3}{7}\\pi $$\n", "\n", "And now we finally have something that we can translate directly into Python, to render our heptagon and heptagrams with full symmetry. Since we defined `drawHeptagrams` to accept a rendering function already, we can just call it again with the new function." ] }, { "cell_type": "code", "execution_count": 17, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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rVgKJgH4+tFV4rhAA4DzQGYwBl/1og2WBBZf74EO/m3CXGCe44G+5M9KqrbC4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374xzTih8cpKdGErjZscEVAAP1CV60xZ449IiIEmDFDYvJ/K6VUiewfsh8KEp7b9BzGnBiD9qM0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\njY4m7OmnF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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "def skewRender(v):\n", " x = float( v[0] )\n", " y = float( v[1] )\n", " x = x + y/(2*HeptagonNumber.sigma_real)\n", " y = math.sin( (3.0/7.0) * math.pi ) * y\n", " return [ x, y ]\n", "\n", "drawHeptagrams( skewRender )" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Summary, and a Look Ahead" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "I've shown you how to use heptagon numbers to represent the vertices of a regular heptagon in a very natural way, and how to draw the result faithfully. There are many more interesting figures that can be drawn using these techniques, including some beautiful aperiodic tilings:\n", "\n", "https://tilings.math.uni-bielefeld.de/substitution/danzers-7-fold/\n", "\n", "https://tilings.math.uni-bielefeld.de/substitution/danzers-7-fold-original/\n", "\n", "http://tilings.math.uni-bielefeld.de/substitution/goodman-strauss-7-fold-rhomb\n", "\n", "To be able to generate figures like those, you'll need to know how to apply the rotations that are at the heart of the symmetry of the heptagon.\n", "\n", "In [Part 3](https://nbviewer.org/github/vorth/ipython/blob/master/heptagons/Sevenfold%20Rotation.ipynb), I will show you how to apply such rotations, and in particular how to render the recursive \"nautilus shell\" figure shown at the beginning of this article." ] } ], "metadata": { "kernelspec": { "display_name": "Python 2", "language": "python", "name": "python2" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 2 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython2", "version": "2.7.10" } }, "nbformat": 4, "nbformat_minor": 0 }