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"content": "<p>A problem mentioned elsewhere; what is the largest similar triangle that can be embedded into a triangle?<br /> i.e 3:4:5.<br />For any triangle, we can determine a lower bound (in fact I think it is sharp).<br />Take \\(x,y,z\\) as the vertices and \\(X,Y,Z\\) as the opposite sides and<br />\\( x_{\\theta},y_{\\theta},z_{\\theta}\\) as the corresponding angles.<br />Example area of the triangle: \\( A_{z}=\\frac{1}{2}\\cdot X\\cdot Y\\cdot\\sin\\left(z_{\\theta}\\right)\\)<br />But \\( A=A_{z}=A_{y}=A_{x}\\)<br />The same for any other vertice order.<br />Join the midpoints of the sides to form vertices \\( x', y', z'\\) lying on the sides \\( X, Y, Z\\ )\\) <br />And \\(𝑋',𝑌',𝑍' \\) the opposite sides of the new triangle.<br />Trim the “outside” triangles; remembering that one only needs to specify 3 parameters to determine the triangle and “similarity” is just a scaling of terms \\(X, Y, Z\\).<br />Taking the example, we have<br />\\(A_{z}'=\\frac{1}{2}\\frac{X}{2}\\cdot\\frac{Y}{2}\\cdot\\sin\\left(z_{\\theta}\\right)=\\frac{1}{4}\\cdot\\left(\\frac{1}{2}\\cdot X\\cdot Y\\cdot\\sin\\left(z_{\\theta}\\right)\\right) \\) <br />and the same for the other three vertices since the angles haven't changed.<br />Similarly for \\(x_{\\theta},y_{\\theta}\\)<br />So the area of the inside triangle is: <br />\\(A-A'_{z}-A'_{x}-A'_{y}=\\frac{1}{4}A \\) <br />So we have that \\(\\frac{1}{4}\\) works but is it maximum? <br />I am going to read<br />Perpendicular Polygons<br /><a href=\"https://www.jstor.org/stable/2322190\" target=\"_blank\" rel=\"nofollow noopener noreferrer\" translate=\"no\"><span class=\"invisible\">https://www.</span><span class=\"\">jstor.org/stable/2322190</span><span class=\"invisible\"></span></a><br />Which looks to be a paywall, but is available for reading (after numerous clicks). <br />Courtesy of our ongoing squabbles about “Intellectual Property” (as if it was a rock in my yard) <br />In any case, this starts out as sort of, Freshman Complex Analytic Geometry and rapidly gets into material that requires thinking</p>",
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"en": "<p>A problem mentioned elsewhere; what is the largest similar triangle that can be embedded into a triangle?<br /> i.e 3:4:5.<br />For any triangle, we can determine a lower bound (in fact I think it is sharp).<br />Take \\(x,y,z\\) as the vertices and \\(X,Y,Z\\) as the opposite sides and<br />\\( x_{\\theta},y_{\\theta},z_{\\theta}\\) as the corresponding angles.<br />Example area of the triangle: \\( A_{z}=\\frac{1}{2}\\cdot X\\cdot Y\\cdot\\sin\\left(z_{\\theta}\\right)\\)<br />But \\( A=A_{z}=A_{y}=A_{x}\\)<br />The same for any other vertice order.<br />Join the midpoints of the sides to form vertices \\( x', y', z'\\) lying on the sides \\( X, Y, Z\\ )\\) <br />And \\(𝑋',𝑌',𝑍' \\) the opposite sides of the new triangle.<br />Trim the “outside” triangles; remembering that one only needs to specify 3 parameters to determine the triangle and “similarity” is just a scaling of terms \\(X, Y, Z\\).<br />Taking the example, we have<br />\\(A_{z}'=\\frac{1}{2}\\frac{X}{2}\\cdot\\frac{Y}{2}\\cdot\\sin\\left(z_{\\theta}\\right)=\\frac{1}{4}\\cdot\\left(\\frac{1}{2}\\cdot X\\cdot Y\\cdot\\sin\\left(z_{\\theta}\\right)\\right) \\) <br />and the same for the other three vertices since the angles haven't changed.<br />Similarly for \\(x_{\\theta},y_{\\theta}\\)<br />So the area of the inside triangle is: <br />\\(A-A'_{z}-A'_{x}-A'_{y}=\\frac{1}{4}A \\) <br />So we have that \\(\\frac{1}{4}\\) works but is it maximum? <br />I am going to read<br />Perpendicular Polygons<br /><a href=\"https://www.jstor.org/stable/2322190\" target=\"_blank\" rel=\"nofollow noopener noreferrer\" translate=\"no\"><span class=\"invisible\">https://www.</span><span class=\"\">jstor.org/stable/2322190</span><span class=\"invisible\"></span></a><br />Which looks to be a paywall, but is available for reading (after numerous clicks). <br />Courtesy of our ongoing squabbles about “Intellectual Property” (as if it was a rock in my yard) <br />In any case, this starts out as sort of, Freshman Complex Analytic Geometry and rapidly gets into material that requires thinking</p>"
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"published": "2022-12-19T15:04:33Z",
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"content": "<p>Introduction to an exposition on some work I have done over the last decade. Since I believe that "motivation" in mathematics is the best motivator 😀 . Explanations, Theorems, etc ... will follow; slowly I am reviewing the documentation (for the 11th time) again.<br />1) In the realm of polynomial generating functions, EGF's (Sheffer sequences <a href=\"https://en.wikipedia.org/wiki/Sheffer_sequence\" target=\"_blank\" rel=\"nofollow noopener noreferrer\" translate=\"no\"><span class=\"invisible\">https://</span><span class=\"ellipsis\">en.wikipedia.org/wiki/Sheffer_</span><span class=\"invisible\">sequence</span></a>), OGF's, and similar sequences are all "similar" when expressed as polynomial coefficient arrays/matrices.<br /><a href=\"https://mathstodon.xyz/tags/EGF_OGF\" class=\"mention hashtag\" rel=\"tag\">#<span>EGF_OGF</span></a><br />1/n</p>",
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"en": "<p>Introduction to an exposition on some work I have done over the last decade. Since I believe that "motivation" in mathematics is the best motivator 😀 . Explanations, Theorems, etc ... will follow; slowly I am reviewing the documentation (for the 11th time) again.<br />1) In the realm of polynomial generating functions, EGF's (Sheffer sequences <a href=\"https://en.wikipedia.org/wiki/Sheffer_sequence\" target=\"_blank\" rel=\"nofollow noopener noreferrer\" translate=\"no\"><span class=\"invisible\">https://</span><span class=\"ellipsis\">en.wikipedia.org/wiki/Sheffer_</span><span class=\"invisible\">sequence</span></a>), OGF's, and similar sequences are all "similar" when expressed as polynomial coefficient arrays/matrices.<br /><a href=\"https://mathstodon.xyz/tags/EGF_OGF\" class=\"mention hashtag\" rel=\"tag\">#<span>EGF_OGF</span></a><br />1/n</p>"
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"content": "<p><a href=\"https://mathstodon.xyz/tags/EGF_OGF\" class=\"mention hashtag\" rel=\"tag\">#<span>EGF_OGF</span></a><br />Of course, \\( [P(r)] \\) could have been applied to the left. Both change the polynomial value. But we can change the basis and keep the polynomial value.<br />\\[ \\left[\\begin{array}{ccccc}<br />a_{0} & a_{1} & a_{2} & \\cdots & a_{n}\\end{array}\\right]\\left[\\begin{array}{c}<br />P(r)\\end{array}\\right]^{-1}\\left[\\begin{array}{c}<br />P(r)\\end{array}\\right]\\left[\\begin{array}{c}<br />1\\\\<br />x\\\\<br />x^{2}\\\\<br />\\cdots\\\\<br />x^{n}<br />\\end{array}\\right] \\] <br />4/n :)</p>",
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"en": "<p><a href=\"https://mathstodon.xyz/tags/EGF_OGF\" class=\"mention hashtag\" rel=\"tag\">#<span>EGF_OGF</span></a><br />Of course, \\( [P(r)] \\) could have been applied to the left. Both change the polynomial value. But we can change the basis and keep the polynomial value.<br />\\[ \\left[\\begin{array}{ccccc}<br />a_{0} & a_{1} & a_{2} & \\cdots & a_{n}\\end{array}\\right]\\left[\\begin{array}{c}<br />P(r)\\end{array}\\right]^{-1}\\left[\\begin{array}{c}<br />P(r)\\end{array}\\right]\\left[\\begin{array}{c}<br />1\\\\<br />x\\\\<br />x^{2}\\\\<br />\\cdots\\\\<br />x^{n}<br />\\end{array}\\right] \\] <br />4/n :)</p>"
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"published": "2022-12-24T14:56:47Z",
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"content": "<p><span class=\"h-card\" translate=\"no\"><a href=\"https://mathstodon.xyz/@rrogers\" class=\"u-url mention\">@<span>rrogers</span></a></span> Just Latex practice:<br />Utility of Pascal's Matrix<br />\\[ \\left[\\begin{array}{ccccc}<br />a_{0} & a_{1} & a_{2} & \\cdots & a_{n}\\end{array}\\right]\\left[\\begin{array}{c}<br />P(r)\\end{array}\\right]\\left[\\begin{array}{c}<br />1\\\\<br />x\\\\<br />x^{2}\\\\<br />\\cdots\\\\<br />x^{n}<br />\\end{array}\\right] \\] <br />\\[ =\\left[\\begin{array}{ccccc}<br />a_{0} & a_{1} & a_{2} & \\cdots & a_{n}\\end{array}\\right]\\left[\\begin{array}{c}<br />1\\\\<br />\\left(x+r\\right)\\\\<br />\\left(x+r\\right)^{2}\\\\<br />\\cdots\\\\<br />\\left(x+r\\right)^{n}<br />\\end{array}\\right] \\] <br /><a href=\"https://mathstodon.xyz/tags/EGF_OGF\" class=\"mention hashtag\" rel=\"tag\">#<span>EGF_OGF</span></a> <br />3/n</p>",
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"en": "<p><span class=\"h-card\" translate=\"no\"><a href=\"https://mathstodon.xyz/@rrogers\" class=\"u-url mention\">@<span>rrogers</span></a></span> Just Latex practice:<br />Utility of Pascal's Matrix<br />\\[ \\left[\\begin{array}{ccccc}<br />a_{0} & a_{1} & a_{2} & \\cdots & a_{n}\\end{array}\\right]\\left[\\begin{array}{c}<br />P(r)\\end{array}\\right]\\left[\\begin{array}{c}<br />1\\\\<br />x\\\\<br />x^{2}\\\\<br />\\cdots\\\\<br />x^{n}<br />\\end{array}\\right] \\] <br />\\[ =\\left[\\begin{array}{ccccc}<br />a_{0} & a_{1} & a_{2} & \\cdots & a_{n}\\end{array}\\right]\\left[\\begin{array}{c}<br />1\\\\<br />\\left(x+r\\right)\\\\<br />\\left(x+r\\right)^{2}\\\\<br />\\cdots\\\\<br />\\left(x+r\\right)^{n}<br />\\end{array}\\right] \\] <br /><a href=\"https://mathstodon.xyz/tags/EGF_OGF\" class=\"mention hashtag\" rel=\"tag\">#<span>EGF_OGF</span></a> <br />3/n</p>"
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