building a basis whilst exploring the ocean ...


  • 2021/05/22 - migrating to obsidian
  • 2016/12/25 - Solving the Hypercake SeriesSolving the Hypercake Series
    In the [[Hypercake Number| previous post]] about hypercake numbers, we obtained a recurrence relation for the hypercake number:

    \[c_{m,n} = c_{m,n-1} + c_{m-1,n-1}\]

    with $c_{m,1} = 2$ and $c_{1,m} = m+1$.

    To move forward, some basic understanding of generating functions1 and binomial coefficients2 is required. For the sake of completeness and simplifying things let us also consider the case of 0 cuts which yield $c_{m,0} = 1$ (max cake slices with $0$ cuts) and $c_{0,n} = 1$ (a point can...
  • 2016/12/18 - Hypercake NumberHypercake Number
    Cake number, as defined in mathematics is the number of maximum pieces a cake can be cut into with a given number of cuts. In this post, we will try to do the same with a cake (a Hypercake if you may) in higher dimensions.

    The Knife's Dimension

    Before we move on to cutting cakes, let's ponder upon the dimension of the knife. What should be the dimension of the knife when cutting a multidimensional cake?

    For a regular cake ($3$D), we use a regular knife (a $2$D plane).
    For a cutting pa...