Plate · I / MMXXVI

Sixty‑Seven

A study in stroke neighbourhoods,
extruded along z.

Fig. I The solid $\mathcal{S}_{3D}$ at $\alpha = -\pi/6$, $\beta = \arctan\!\bigl(\tfrac{1}{\sqrt{2}}\bigr)$. drag to orbit · wheel to zoom

§ III Digit 6

Loop₆

closed annular stroke

$(a,\, b,\, r) \;=\; (-0.55,\; -0.40,\; 0.34)$

Full circle about $(a,b)$. The sub-level set $\{d_\circ \le T\}$ is the annulus of inner radius $r{-}T$ and outer $r{+}T$.

Hook₆

quarter arc · top → left

$(a, b, r, \theta_1, \theta_2) = \bigl({-}0.50,\; {-}0.05,\; 0.42,\; \tfrac{\pi}{2},\; \pi\bigr)$

Angular support $[\pi/2,\pi]$. Off-support, the stroke collapses smoothly onto the nearer endpoint $q_1$ or $q_2$.

§ IV Digit 7

Bar₇

segment · upper cross-stroke

$A = (0.15,\; 0.55), \;\; B = (0.95,\; 0.55)$

Horizontal; shares its right endpoint with Diag₇, so the two strokes are connected.

Diag₇

segment · descending

$A = (0.95,\; 0.55), \;\; B = (0.35,\; -0.65)$

Slope $-2$; it sweeps from the top-right corner of $\mathcal{D}_7$ down toward the lower interior.

Stroke & extrusion

T 0.065

h 0.300

Projection

α −30.000°

β 35.264°

Composition

§ I Stroke neighbourhood

For a curve $\gamma \subset \mathbb{R}^2$ with Euclidean distance $d_\gamma(p) = \inf_{q\in\gamma}\lVert p - q \rVert$, the stroke of half-width $T$ is the sub-level set $S_T(\gamma) = \{\, p \in \mathbb{R}^2 \,:\, d_\gamma(p) \le T \,\}$. We fix $T = 0.065$.

§ II Primitives

$d_\circ(p;\,a,b,r) \;=\; \bigl|\sqrt{(x-a)^2 + (y-b)^2}\;-\;r\bigr|$

$d_\frown(p;\,a,b,r,\theta_1,\theta_2) = \begin{cases}\bigl|\sqrt{(x-a)^2+(y-b)^2}-r\bigr|, & \varphi(p)\in[\theta_1,\theta_2]\\[4pt]\min\{\,\lVert p-q_1\rVert,\;\lVert p-q_2\rVert\,\}, & \text{otherwise}\end{cases}$

$d_\ell(p;\,A,B) = \bigl\lVert\,p - \bigl(A + \mathrm{clip}_{[0,1]}(t^*(p))\,(B-A)\bigr)\bigr\rVert,\;\; t^*(p) = \tfrac{(p-A)\cdot(B-A)}{\lVert B-A\rVert^2}$

§ VI Extrusion

For $h = 0.30$, $\mathcal{S}_{3D} = \mathcal{S}_{2D} \times [0, h]$ — the right cylinder of the planar figure.

§ VII Isometric projection

$\begin{pmatrix}u\\v\end{pmatrix} = \begin{pmatrix}-\sin\alpha & \cos\alpha & 0\\[2pt]-\cos\alpha\sin\beta & -\sin\alpha\sin\beta & \cos\beta\end{pmatrix}\!\begin{pmatrix}x\\y\\z\end{pmatrix}$

True isometry requires $\beta = \arctan(1/\sqrt{2})$, so the three axes project to lines $120°$ apart — a unit cube becomes a regular hexagon. The preview uses $\alpha = -\pi/6$, a slight dimetric rotation exposing the interior of $\mathcal{D}_7$.

§ VIII Summary

$\text{“67”} \;=\; \bigl(\,S_T(\text{Loop}_6) \cup S_T(\text{Hook}_6)\,\bigr) \;\cup\; \bigl(\,S_T(\text{Bar}_7) \cup S_T(\text{Diag}_7)\,\bigr) \;\times\; [0, h]$