In describing my work, I’ve encountered some people have a hard time understanding graphs in a phase plane, where I plot one physical parameter against another. I use these graphs to show how two variables interact. I want to look at data in a time free manner like this because if I were to examine real patients I wouldn’t know when they got infected. This made me look for a literal-minded compromise that would let me show both a time series and a phase plane.
I could, and do, make 3D graphs on my computer that show the two physical parameters as well as time, but to show these I need my computer; drawing these graphs in the air with my finger leads to blank stares in my audience. Personally, I find it hard to understand a 3D image on a screen without rotating the graph myself and I worry that when I show someone else one of these figures on a computer that they share this problem.
To let people manipulate data on their own, I’ve tried printing 3D graphs in plastic. In theory I can carry these around in my pocket and show them whenever the opportunity arises and my audience can examine the data themselves. In the picture below I show what happens to a mouse infected with malaria by tracking changes in red blood cells and reticulocytes over time. The mice start in the green area where they are symptom free. Malaria makes the mouse anemic and RBCs drop in the blue painted part of the curve. The RBCs recover to normal levels in the yellow part. Reticulocytes are replacement red blood cells; they dip slightly before rising to high levels to replace the damaged red blood cells. If you look at the graph end on you can see the path the animals trace through an RBC by reticulocyte phase plane. First the RBCs drop and then the reticulocytes rise to replace the RBCs and eventually the system returns to where it started. One nice thing about this view is that it is clear that the host is in the same physiological state at the beginning and end of the infection; if you draw a timeline, the beginning and end are at opposite sides of the graph and it is less obvious that the start and finish are identical.
To print these graphs, I made a 3D model of the data using Autocad. I traced a polyline that used reticulocytes and RBCs with a scale of zero to one with time plotted along the X axis. I swept a circle along the polyline to give the graph volume. To give the graph perspective, I included some walls that also help protect the trace. I printed these using fused filament fabrication with ABS plastic that I covered with acrylic paint. The prints are fragile as the graph hangs free and it isn’t easy to change this as I want it to be free so that I can show it end on to see the phase plane. I find these graphs useful in my office when I’m explaining a project but I don’t carry them around with me. I’ve given talks where I hand out a shoebox of these to the audience but haven’t seen any enough “Aha!” moments that make me want to keep dragging that box around.