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Active Inference Project

Andrius: I am studying how triangle centers manifest a language by which meaning arises. I am analyzing examples from the Encyclopedia of Triangle Centers. I want to model that with active inference.

One idea for modeling the unconsciously answering mind (by which issues come to matter) and the consciously questioning mind (by which meaning arises) is to study how they interact in studying a particular triangle (with specified lengths of sides and sizes of angles) which the answering mind operates on, while the questioning minds refers to general concepts applying to all triangles. Error arises because while the answering mind executes perfectly, yet its observations have a margin for error. Now for particular triangles there may be results (namely, coinciding triangle centers) which are not true of triangles in general. This gives rise to meaningful statements about particular triangles. These statements can be suggested by the answering mind. But they need to be verified and established by the questioning mind. This can be done interactively or this process can be guided by the third investigatory mind of consciousness. My thought is to model this with Active Inference.

Triangle Centers

Vocabulary

• The Encyclopedia of Triangle Centers has a glossary related tables, and a collection of links
• See also: Mathworld: Triangle and Triangle, List of triangle topics, Encyclopedia of Triangle Centers.

Analysis of Triangle Centers

A triangle center is defined as the concurrent intersection of three lines which are defined in terms of the intrinsic properties of the triangle (notably the relative lengths of sides, but independent of the triangle's placement or scale or context). This can be thought of as the three minds coming together to focus on a single point. Triangle centers form a multiplicative group for which the incenter is the identity.

ID and Name	Definition	Set of concepts	Interpretation	Notes
X(1) incenter	A) Equidistant from the three sides. B) Internal angle bisectors intersect.	A) Geodesics (of equal length), points of intersection, arcs of circle. B) Bisected angles.	A) Place a circle inside. Expand it to its maximum size. B) Fold the angles to bisect them. Intersect the resulting lines.	Why do these yield the same center?
X(2) centroid	A) Lines from points to medians. B) Arithmetic mean position of all of the points on the triangle	A) Medians (bisected sides) B) Weight of sides	A) Fold each side to create medians. Draw lines from points to medians. B) Build the triangle from three lines with mass. Discover the center of mass	Why do these yield the same center?
X(3) circumcenter	A) Center of the circle including the three vertices. B) Intersector of the perpendicular bisectors of the sides	A) Circle and arcs B) Lines and right angles	A) Place circle around the triangle. Shrink the circle to its minimum size. B) Bisect the sides. Draw perpendicular lines at the bisectors.	Why do these yield the same center?
X(4) orthocenter	intersection of the three altitudes		A) draw lines perpendicular to each line, slide each to a vertex equivalently: draw a cross, align a side, draw a line down from the vertex
X(5) nine-point center	center of the nine-point circle, which includes the midpoint of each side, the foot of each altitude, and the midpoint from each vertex to the orthocenter		given any three of the nine points, all distinct, find the circumcenter X(3)
X(6) symmedian point	intersection of the three symmedians (which reflect a median across the angle bisector)	median, angle bisector, reflection, intersection with side	fold lines to get medians, fold angles to get bisectors, fold lines over bisectors to get symmedians from medians
X(7) Gergonne point	A) intersection of lines from vertices to touch points of the incircle B) symmedian point of triangle of touch points		A) Given X(1), draw lines from the vertices to the touch points B) If the vertices are not available, given the touch points, build X(6)
X(8) Nagel point	intersection of lines from each vertex to the corresponding semiperimeter point
X(9) Mittenpunkt	symmedian point of the triangle formed by the centers of the three excircles
X(10) Spieker center	center of the Spieker circle
X(11) Feuerbach point
X(13) Fermat point
X(15) first isodynamic point
X(16) second isodynamic point
X(17) first Napoleon point
X(18) second Napoleon point
X(19) Clawson point
X(20) de Longchamps point
X(21) Schiffler point
X(22) Exeter point
X(39) Brocard midpoint
X(40) Bevan point
X(175) Isoperimetric point
X(176) Equal detour point