# VectorPaint two-round triangle removal: check four holes and nine retained triangles

Language: en
Canonical: [VectorPaint two-round triangle removal: check four holes and nine retained triangles](<https://allluckly.cn/en/guides/vectorpaint-two-round-sierpinski-triangles/>)
中文：[中文 HTML](<https://allluckly.cn/guides/vectorpaint-two-round-sierpinski-triangles/>)
English: [English HTML](<https://allluckly.cn/en/guides/vectorpaint-two-round-sierpinski-triangles/>)
Published: 2026-11-10T00:00:00+08:00
Modified: 2026-11-10T09:39:21+08:00
Author: Bison

Remove central midpoint triangles in two rounds,checking holes,retained pieces and area fractions separately without automatic fractal generation.

English translation of the [Chinese original](<https://allluckly.cn/guides/vectorpaint-two-round-sierpinski-triangles/>), translated on 2026-11-10. Original publication dates, attribution, code and version notes are retained.

Hole count, retained-triangle count and area fraction differ in recursive patterns. Make a finite two-round Sierpiński-style construction, preserving round 1 before operating on its three retained triangles. Build shapes manually without a fractal tool. Two rounds are not the infinite limit or a fabrication template.

## Preserve one complete equilateral triangle

Use base endpoints(200, 800), (1000, 800), apex(600, 800−400√3), approximately y 107.18, side 800. Rounded placement differs from theoretical coordinates. Draw a complete closed triangle labelled round 0, not an already-removed first round. Mark midpoints faintly with external labels. These are exercise dimensions, not natural anatomy. Duplicate before each round and retain the baseline so later holes remain explainable.

## Join midpoints and remove the first centre

Connecting side midpoints yields four equal-area triangles of side 400. Remove the central inverted one, leaving three corner pieces: one hole, three retained pieces, area 3/4. A background-coloured patch is visual, not necessarily transparent, and must be rechecked on another background. Build three retained triangles directly or use available masking without assuming fractal/difference commands. Verify midpoints and dimensions rather than carving an arbitrary central triangle.

## Apply round 2 separately to each retained piece

Within each of the three round 1 pieces find side midpoints and remove a central side 200 triangle. Each parent leaves three corners, giving nine retained side 200 triangles. Do not operate again inside the original hole or process only the top parent and call round 2 complete. Label parentsA/B/C and check three operations. Each new hole belongs to its parent’s orientation; arbitrary canvas-centred scaling is not a substitute. All smaller pieces follow one rule.

## Distinguish four holes from nine retained pieces

The original large hole remains and three new holes give four, not nine. Large-hole area is 1/4 and each new hole 1/16, total 7/16; retained area is 9/16. Nine retained pieces also each occupy 1/16, providing an independent check. Count empty versus filled regions separately. Point contacts do not establish manufacturable or load-bearing material continuity; no such physical inference is made.

## Multiply retained area by three quarters per round

Retained fractions are 1, 3/4, 9/16, not 1/2 from repeatedly subtracting 1/4 of the original. Round 2 removes 1/4 of the remaining three pieces, or 3/16 of the original. Compare multiplication with hole subtraction and check parent scope for disagreement. Pixel-area measurement is unnecessary; ideal area excludes strokes. Theoretical fractions are not automatic measurements of antialiased output. [Cambridge NRICH’s Sierpinski-triangle exercise](<https://nrich.maths.org/problems/sierpinski-triangle>) supports retaining three quarters per round; this drawing stops after two finite rounds and adopts no infinite-process limit conclusion.

## Check round completeness with a partial-operation copy

In a wrong copy apply round 2 only to the top parent, leaving lower parents at round 1. It has the large hole plus one new hole, not a complete round 2, and unequal retained sizes. Identify unprocessed parents rather than making the title vague. Preserve the complete version; restore all three operations and recount four/nine. A third round would require all nine parents and a new label, not an assertion that more detail completes an infinite fractal.

## Check whether output holes really expose background

Save and inspect PNG on light/dark backgrounds for four holes, nine pieces and the original outline. Persistent white triangles reveal colour patches rather than transparency; label or rebuild as required instead of blaming export. Reduce strokes or enlarge globally for merging pieces rather than deleting one for space. Hide unwanted midpoint/parent marks actively. Compare rounds at one scale so relative hole sizes retain meaning.

## Recheck parents, rounds, counts and area

For missing holes inspect parents; for unequal nine pieces inspect midpoints; for 1/2 area inspect the current-area rule. Repair one parent and recount rather than scaling all pieces and breaking joins. Retain rounds 0/1/2, counterexample and output with reproducible empty/filled counts. This explains finite recursion, not automatic fractal generation, infinite-limit verification or physical cutting integrity.

## Related tutorials and sources

[VectorPaint seven-piece puzzles: fix the pieces before checking a silhouette](<https://allluckly.cn/en/guides/vectorpaint-seven-piece-puzzle/>), [VectorPaint area comparisons: calculate side lengths before arranging 1, 4 and 9 units](<https://allluckly.cn/en/guides/vectorpaint-area-comparison/>).

Checked on November 10, 2026: VectorPaint public version 1.0.8, requiring iOS / iPadOS 17.0 or later. Capability scope follows the [official App Store listing](<https://apps.apple.com/cn/app/id1587272880>) and [Apple public version lookup](<https://itunes.apple.com/lookup?id=1587272880&country=cn>). This is an original exercise to perform independently; use controls in your installed version. Manual construction and external records do not imply new automatic analysis features. No step-by-step physical-device, private-file or all-output-environment testing is claimed.