Shape regularity and local reduction under 2D newest-vertex bisection

Write the parent vertices as $A,B,C$, with reference edge $AB$, midpoint $M$, and squared side lengths $X=\mathrm{\mid }AC{\mathrm{\mid }}^2$, $Y=\mathrm{\mid }BC{\mathrm{\mid }}^2$, $Z=\mathrm{\mid }AB{\mathrm{\mid }}^2$. The two first children have edge-length squares $\{X,Z\mathrm{/}4,W\}$ and $\{Y,Z\mathrm{/}4,W\}$, where $W=\mathrm{\mid }(A+B)\mathrm{/}2-C{\mathrm{\mid }}^2=(2X+2Y-Z)\mathrm{/}4$ by expanding the squared norm. The four labeled types have squared edge lists $\{X,Y,Z\}$, $\{Z\mathrm{/}4,X,W\}$, $\{Z\mathrm{/}4,Y,W\}$, and $\{X\mathrm{/}4,Y\mathrm{/}4,W\}$. In type 1 the reference side has square $X$, and the new median square is $(2(Z\mathrm{/}4)+2W-X)\mathrm{/}4=Y\mathrm{/}4$; its children therefore have side lists $\{X\mathrm{/}4,Z\mathrm{/}4,Y\mathrm{/}4\}$ and $\{X\mathrm{/}4,W,Y\mathrm{/}4\}$, which are a quarter-scale type 0 and type 3. The type 2 calculation interchanges $X,Y$. In type 3 the reference side has square $W$, and its median square is $(2(X\mathrm{/}4)+2(Y\mathrm{/}4)-W)\mathrm{/}4=Z\mathrm{/}16$; its children are quarter-scale types 1 and 2. Type 0 gives types 1 and 2 by the first bisection. Hence this finite list is closed under every labeled bisection and no other normalized shape occurs.

Each of the finitely many normalized triangles in step 1.1 is nondegenerate: its area is half the parent area at each bisection and its side lengths are the explicit midpoint lengths, so its three angles are positive. The minimum of these finitely many angles and the initial minimum angle is positive. Hence all descendants have a common positive minimum angle and a common shape-regularity bound. The angle minimum over this finite list is positive; the maximum shape-regularity constant over it and the initial cells is therefore finite.

After the first generation the midpoint $M$ of $AB$ is present. The second generation bisects $AC$ and $BC$, adding their midpoints $N$ and $P$. The four second-generation triangles have reference edges on the segments $AM,BM,CM$; the third generation bisects those segments, in particular $CM$, adding its midpoint $S$ in the interior of $ABC$. The eight leaves have vertex triples $(A,N,Q),(Q,M,N),(N,M,S),(N,S,C),(P,M,R),(P,R,B),(P,C,S),(P,S,M)$, where $Q=(A+M)\mathrm{/}2$ and $R=(B+M)\mathrm{/}2$. Subtracting these vertex coordinates shows that each leaf side is an original side divided by $2$, the median $\mathrm{\mid }C-M\mathrm{\mid }$ divided by $2$, or the reference side divided by $4$. Since $M$ lies on $AB$, convexity of the norm gives $\mathrm{\mid }C-M\mathrm{\mid }\le \max \{\mathrm{\mid }C-A\mathrm{\mid },\mathrm{\mid }C-B\mathrm{\mid }\}\le \mathrm{diam}(ABC)$. Thus every leaf side, and hence every leaf diameter, is at most half the parent diameter.