So that's how it works for four sides. But for a pentagon, a five-sided shape, it turns out you need nine dots. For a hexagon, it's 17 dots. But beyond that, we don't know. It's a mystery how many dots is required to create a heptagon or any larger shapes. More importantly, there should be a formula to tell us how many dots are required for any shape. Mathematicians suspect the equation is M=1+2 N-2 , where M is the number of dots and N is the number of sides in the shape. But as yet, they've only been able to prove that the answer is at least as big as the answer you get that way.
Twenty-six adults were given a chair massage and 24 control group adults were asked to relax in the massage chair for 15 minutes, two times per week for five weeks. On the first and last days of the study they were monitored for EEG, before, during and after the sessions. In addition, before and after the sessions they performed math computations, they completed POMS Depression and State Anxiety Scales and they provided a saliva sample for cortisol. At the beginning of the sessions they completed Life Events, Job Stress and Chronic POMS Depression Scales. Group by repeated measures and post hoc analyses revealed the following: 1) frontal delta power increased for both groups, suggesting relaxation; 2) the massage group showed decreased frontal alpha and beta power (suggesting enhanced alertness); while the control group showed increased alpha and beta power; 3) the massage group showed increased speed and accuracy on math computations while the control group did not change; 4) anxiety levels were lower following the massage but not the control sessions, although mood state was less depressed following both the massage and control sessions; 5) salivary cortisol levels were lower following the massage but not the control sessions but only on the first day; and 6) at the end of the 5 week period depression scores were lower for both groups but job stress score were lower only for the massage group.