Apophenia is a computer program written in Microsoft Visual Basic 2008 Express. It is designed to run on PCs running Microsoft Windows XP with SP2 installed and Microsoft Windows Vista.

The program creates and displays graphical plots of groups of positive natural integers arranged in concentric regular polygon rings. By highlighting the prime numbers it is possible for the user to see apparent patterns in the arrangement of the prime and composite numbers irrespective of the order of the polygon.

The prime numbers that lie upon the "lines" of each pattern differ from plot to plot but are generally explained by reference to quadratic equations applied to the ring index, r.

Apophenia also allows the user to highlight the prime and composite results of such quadratic equations.

Apophenia will be released shortly. You can contact us here.

The following pictures illustrate the kinds of patterns which the user can expect to see using Apophenia. Apophenia can save any plot as a .PNG, .BMP, .GIF or .JPG file with complete control over the size, colours, scale and point size. All of the plots on this website were created using Apophenia:

For some more colourful examples of Apophenia plots, including an animated GIF file made from a series of Ulam spiral plots, click here.

Apophenia allows fixed rotations to each ring in a plot. For example, Ulam's Spiral can be plotted by applying a one point anti-clockwise rotation to each ring so that the highest number in each ring is adjacent to the lowest number in the next ring, essential for the spiral appearance of the plot:

Apophenia can also output raw data relating to each plot, including the X and Y coordinates of each plotted point, as a text file in .CSV format, (Comma Separated Variables). CSV format files can be opened for analysis using spreadsheet applications such as Microsoft Excel although limitations in the number of rows that can be displayed mean that you may need to open the .CSV file using a text editor in order to see all the data. The user may select which data to output.

A CSV file generated from the above plot, showing the coordinates of each of the 641 prime numbers and the ring offset value for each ring, can be viewed by clicking here. It is a trivial matter to use Microsoft Excel to open a CSV file and produce an XY graph reproducing the Apophenia plot from which the CSV file was derived.