On the density of shapes in three-dimensional affine subdivision

Qianghua Luo; Jieyan Wang; Qianghua Luo; Jieyan Wang

doi:10.3934/math.2020345

AIMS Mathematics

2020, Volume 5, Issue 5: 5381-5388. doi: 10.3934/math.2020345

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Research article

On the density of shapes in three-dimensional affine subdivision

Qianghua Luo ^1,2,
Jieyan Wang ^{1,2
,
,}

1.
School of Mathematics, Hunan University, Changsha 410082, P. R. China
2.
Hunan Prov key Lab of Intelligent Information Processing and Applied Mathematics, Hunan University, Changsha 410082, P. R. China

Received: 03 April 2020 Accepted: 15 June 2020 Published: 22 June 2020
MSC : 51M20, 52B10

The affine subdivision of a simplex $\Delta$ is a certain collection of $(n+1)!$ smaller $n$-simplices whose union is $\Delta$. Barycentric subdivision is a well know example of affine subdivision(see). Richard Schwartz(2003) proved that the infinite process of iterated barycentric subdivision on a tetrahedron produces a dense set of shapes of smaller tetrahedra. We prove that the infinite iteration of several kinds of affine subdivision on a tetrahedron produce dense sets of shapes of smaller tetrahedra, respectively.
- barycentric subdivision,
- affine subdivision,
- dense set,
- tetrahedra,
- simplex
Citation: Qianghua Luo, Jieyan Wang. On the density of shapes in three-dimensional affine subdivision[J]. AIMS Mathematics, 2020, 5(5): 5381-5388. doi: 10.3934/math.2020345

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Abstract

The affine subdivision of a simplex $\Delta$ is a certain collection of $(n+1)!$ smaller $n$-simplices whose union is $\Delta$. Barycentric subdivision is a well know example of affine subdivision(see). Richard Schwartz(2003) proved that the infinite process of iterated barycentric subdivision on a tetrahedron produces a dense set of shapes of smaller tetrahedra. We prove that the infinite iteration of several kinds of affine subdivision on a tetrahedron produce dense sets of shapes of smaller tetrahedra, respectively.

References

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