Different modes of growth. Cycle halving.
I have highlighted in the table below the different modes of growth including linear, saturation, exponential, and singularity. For singularity growth, singularity is chosen to be x=5. I show progression from x=0 to x=5 of the 4 growth methods. I slow down before x=5 and use smaller cycles to highlight the rapid growth of singularity growth. Notice how singularity growth goes well past all growth models in the last 3% of the studied interval. Ray Kurzweil of Google proposes that technological achievement and economic growth goes by singularity growth. This comes from the theory of cycle halving. As you half cycle length, you encounter an infinite amount of growth cycles to reach a fixed point. That fixed point is singularity. At singularity we reach infinity in progress and cycles are infinitesimally small. An example, if you start with a cycle length 1 and keep halving like .5, .25, .125. The cumulative addition of the cycles reaches 2 after infinite cycles. It takes infinite cycles to reach 2. Each cycle keeps moving the cumulative value closer to 2 but never beyond or completely to 2. I have shown the cycle halving in the second table below.
I wanted to explain more about singularity growth. If you look at economic growth 10,000 years ago it was 3% a century or lower probably. Now it is 3% a year or compounded above 1000% over the last century. Population growth is even behaving like a singularity if you smooth it out over a longer moving average. Natural forces like gravity and electromagnetic forces behave like a singularity. Gravitational and electromagnetic forces go to infinity as distance goes to zero because it is divided by distance. In practice it can’t really go to infinity because you can’t get the distance to zero. But you can go “close to” infinity as distance goes to zero. When you touch your hand on the table, the material really isn’t touching. In reality atoms are vastly free space. The opposite force is from the repulsion of the positively charged nuclei. One thing to note when you have two opposite charges. Potential energy approaches infinity as distance approaches infinity. When you have two same charges potential energy approaches infinite as distance approaches 0. So there is singularities in both potential energy and force. Anyways I propose that my spirit is growing towards infinity at a fixed point in a singularity pace. This is because it is obvious from recent years that there is accelerating geometric growth.