Perception

In a triangle, the point the drawing marks is the incentre, where the three lines of the triangle’s skeleton meet. The centre of mass of a triangle, its centroid, is a different point, and for tall or lopsided triangles the two are far enough apart to tell by eye. Psychologists have studied where people place the “centre” of shapes and have found that attention and taps follow the skeleton, but in the studies found here the incentre was never offered as a candidate. This page sets out what is known, lets you try the task on yourself, and proposes the experiment. It then looks at the other cultural neighbour of the drawing, Islamic geometric pattern, and gives an honest verdict.

1What is known

The medial axis, or skeleton, of a shape is the set of points with more than one nearest point on the boundary. For a triangle it is the three angle bisectors, meeting at the incentre (see the first page). The incentre is also the centre of the largest circle that fits inside, the point the cartographic literature calls the pole of inaccessibility, and the “visual centre” used in some map and interface code.KNOWN

• Kovács and Julesz found contrast enhancement at medial points of shapes (Nature 1994; Vision Research 1998).KNOWN
• Firestone and Scholl (2014) and Ayzenberg and others (2019) report that where people tap on a shape clusters on its skeleton.KNOWN These two are cited from abstracts and summaries: the full pages could not be opened, so the details are unverified.
• Dehaene, Izard, Pica and Spelke (Science 311:381, 2006) found that Munduruku adults and children from age 6 show core geometric intuitions, with a mean of 66.8% on odd-one-out geometry items. Untrained geometric intuition is therefore expected. Nobody is known to have tested incentres there.KNOWN
• On perceived centres of polygons, studies find the perceived centre nearer the area barycentre than other classical centres (the Sofia and Vicario line of work; Yakimoff and others for quadrangles). Proffitt (1982) found the boundary contour dominates. Gatti (1924) reported an upward displacement for equilateral triangles, known here only second-hand. These tested a few isosceles triangles and, as far as the abstracts show, never included the incentre as a candidate.KNOWN (abstract-level reading)

The structural link of the drawing is simple. The three lines drawn from the dot to the corners of its triangle are literally its medial axis.KNOWN (standard fact; checked in research/culture) Whether people who draw them are tracing the skeleton the visual system already computes is a guess.CONJECTURE

2How far apart are the incentre and the centroid?

That depends on the triangle. Measured as the distance between the two points as a percentage of the perimeter:VERIFIED NUMERICALLY (research/culture/centres.py; the last three rows were recomputed for this page)

The drawing’s own right isosceles triangle separates the two points by only 1.7%, which makes it a poor test. An experiment needs tall or lopsided triangles.

3Try it

You will see five triangles one at a time. Tap where you feel the middle of each is. You can tap again to move your mark, and press Next when you are satisfied. After the fifth, the page shows the incentre, the centroid and the skeleton for each, together with your taps. This is a demonstration on one person, with no controls, so it shows nothing about people in general. Nothing leaves this page: it makes no network requests and keeps your taps in this tab’s memory only.

4The proposed experiment

Participants click “the middle” of triangles. The candidate points are the centroid, the incentre, the circumcentre, the orthocentre and the centre of the bounding box. The stimuli must be triangles where the candidates separate, such as the tall and lopsided ones above. Compare two tasks, tapping anywhere (as in Firestone and Scholl) and clicking “the centre”. Compare children aged 5 to 10 with adults, and a non-verbal framing in the style of the Munduruku work (“where would you hide a coin so that it is equally far from all sides?” against “where is the balance point?”). About 60 participants per cell, online, would be cheap.

Predictions: the centre-click tracks the centroid in adults for solid shapes (as in prior work); tap-anywhere shifts toward the incentre and the skeleton; a physical balance instruction moves responses to the centroid.CONJECTURE No study of this form was found in the search.NO REFERENCE FOUND (web searches only) A positive incentre finding would be new.

5Islamic geometric pattern

Islamic star patterns are often built by a method whose modern form is due to Hankin (“polygons in contact”, 1925) and was formalised by Kaplan (2005): from the midpoint of each edge of a tiling, two rays leave at a chosen contact angle into the two neighbouring polygons, and each ray is cut where it meets the ray from the adjacent edge.KNOWN The figure below applies this rule live to the 4.8.8 tiling the drawing makes, and to the tiling of 45° triangles (the square with its X) that sits under the drawing.

Observations from the sweep: on 4.8.8 near 45° the rays give cross-shaped motifs around the squares with the octagons left as voids; from about 67.5° to 78° an 8-point star-shaped void appears in each octagon with four-petal motifs in the squares. On the 45° triangles, a contact angle of 67.5° gives isolated regular octagons, probably an artefact of the simple adjacent-ray cut used.

The verdict. The drawing is not a Hankin construction. Hankin rays start at edge midpoints and are set by an angle; the drawing’s lines start at the incentre and go to the corners, as bisectors. It is a centre-to-vertex subdivision of a tiling, nearer a triangulation or mirror-line drawing than a star pattern. As far as the search found, it is not a known traditional construction. That is absence of evidence only: the searches were limited and a key paper (Kaplan 2005) could not be parsed.NO REFERENCE FOUND I could not confirm that any frame of the sweep is a specific named historical pattern (girih, Mamluk, jali or kumiko). Matching them properly would need Kaplan’s gallery or Sakkal’s corpus.

Three structural observations about the drawing, by construction rather than from history.VERIFIED NUMERICALLY First, the square with its X alone is the tetrakis square tiling, the dual (the shape-swapped partner) of 4.8.8. So the drawing is the tetrakis tiling with its own dual laid over it, the dual’s vertices at the triangle incentres. The dual is a standard fact; the incentre is the choice that makes 4.8.8 regular. Second, the octagon’s inradius is 1, half the side of the square, so the octagons are the tangent unit circles at the square’s corners. Circle-packing-driven star rosettes exist (Brewer, Zha, Neno and Bridges, 2022), the closest neighbour in the literature.KNOWN Third, at the centre of each octagon you see 16 rays: the 8 axes and the 8 vertex spokes, the full set of mirrors of a regular octagon, though the tiling’s own symmetry group has only 4 mirror directions. The drawing carries a local 8-fold rosette that the lattice does not. The framing is mine.

In 3D, Islamic ornament is the muqarnas, and lifting 2D tessellations to muqarnas by algorithm is still open (Bridges 2025).KNOWN No literature was found on space-filling 3D star lattices from Wythoff points. A 3D contact rule on the truncated cubic honeycomb looks untried, but it was not searched exhaustively.SPECULATIVE

6As a path for teaching

Kaleidoscope and Wythoff teaching exists (Palmer, Bridges 2011; the Bridges 2015 kaleidoscope workshop; Freire’s vZome Wythoff page).KNOWN No curriculum was found that starts from “put a dot at the incentre of each triangle of a square” and ends at Wythoff points and crystal structures. A possible path: draw the figure; find 4.8.8 in it; ask why the dot is tan 22.5° up; meet mirror lines and symmetry groups; repeat in 3D and meet the radius ratios 0.225, 0.414 and 0.732; finish with Hankin rays as the art side.SPECULATIVE A workshop paper at Bridges would fit.