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<h1>The Christmas-Tree Sizing Theorem</h1>
<span class="meta">Posted by
<a href="/author/marco-santoni.html">Marco Santoni</a>
on Sun 02 December 2018
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<p>How many Christmas balls should I buy for my tree?</p>
<blockquote>
<p><strong>Theorem.</strong> Let <span class="math">\(H\)</span> be the height of the leaf-cone excluding the trunk part of the tree. Let <span class="math">\(B\)</span> be the diameter of the base of the leaf-cone. The number of the Christmas balls to hang on the tree is given by</p>
</blockquote>
<div class="math">$$
a=b
$$</div>
<p><strong>Proof.</strong> We assume that the balls are hanged on the tree along a grid at regular intervals. Let <span class="math">\(\Delta h\)</span> be the horizontal distance between two consecutive balls in a row; let <span class="math">\(\Delta v\)</span> be the vertical distance between two consecutive rows of balls.</p>
<p>The rows along the cone are defined by <span class="math">\(i \in \{1, 2, \dots, \frac{H}{\Delta v} \}\)</span></p>
<p>Let <span class="math">\(A\)</span></p>
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