Tetrahedra with Integer Edges and Integer Volume

Dove, Kevin L.; Sumner, John S.

doi:10.1080/0025570x.1992.11995991

Cited by 5 publications

(5 citation statements)

References 2 publications

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“…Distances between the Cc of Asp-350, the Nd1 of His-470, the Cc of Asp-472 and the Sc of Cys-646, the four major amino acids of the active site, were computed using ''SPDBV''. From these distances, the volume of the tetrahedrons formed between these four reactive atoms was obtained according to the Euler formula [54] (electronic supplementary material, ESM, Fig. S1).…”

Section: Discussionmentioning

confidence: 99%

Deimination is regulated at multiple levels including auto-deimination of peptidylarginine deiminases

Méchin

Coudane

Adoue

et al. 2010

Cell. Mol. Life Sci.

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Peptidylarginine deiminases (PADs) catalyze deimination, converting arginyl to citrullyl residues. Only three PAD isotypes are detected in the epidermis where they play a crucial role, targeting filaggrin, a key actor for the tissue hydration and barrier functions. Their expression and activation depends on the keratinocyte differentiation state. To investigate this regulation, we used primary keratinocytes induced to differentiate either by increasing cell-density or by treatment with vitamin D. High cell-density increased PAD1 and 3, but not PAD2, at the mRNA and protein levels, and up-regulated protein deimination. By contrast, vitamin D increased PAD1-3 mRNA amounts, with distinct kinetics, but neither the proteins nor the deimination rate. Furthermore, auto-deimination was shown to decrease PAD activity, increasing the distances between the four major amino acids of the active site. In summary, deimination can be regulated at multiple levels: transcription of the PADI genes, translation of the corresponding mRNAs, and auto-deimination of PADs.

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Section: Discussionmentioning

confidence: 99%

Deimination is regulated at multiple levels including auto-deimination of peptidylarginine deiminases

Méchin

Coudane

Adoue

et al. 2010

Cell. Mol. Life Sci.

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“…This is a clear difference from the situation for Heron triangles, although we show in Section 3 that if a tetrahedron is Heron with integer edges then it has integer volume and face areas. Dove and Sumner [10] note that a mod 3 search gives (12V ) 2 ≡ 0 or 1, but they want integer volume and so they just discard the cases where (12V ) 2 ≡ 1 (mod 3). Sierpiński [25] mentions the tetrahedron (3,3,4,3,3,4), which has V = 8 3 , but does not comment on the fact that the volume is not an integer, and Buchholz [4] lists the tetrahedron shown in Fig.…”

Section: Some Historymentioning

confidence: 98%

“…Other rational tetrahedra with rational volume include (7,12,15,12,9,8) [10] showed that if a tetrahedron has integer edges and integer volume, then the volume is divisible by 3. For every multiple of 3 up to 99 (except 87) they found a tetrahedron with that volume.…”

Section: Some Historymentioning

confidence: 98%

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Rational and Heron tetrahedra

Chisholm

MacDougall

2006

Journal of Number Theory

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Buchholz [R.H. Buchholz, Perfect pyramids, Bull. Austral. Math. Soc. 45 (1991) 353-368] began a systematic search for tetrahedra having integer edges and volume by restricting his attention to those with two or three different edge lengths. Of the fifteen configurations identified for such tetrahedra, Buchholz leaves six unsolved. In this paper we examine these remaining cases for integer volume, completely solving all but one of them. Buchholz also considered Heron tetrahedra, which are tetrahedra with integral edges, faces and volume. Buchholz described an infinite family of Heron tetrahedra for one of the configurations. Another of the cases yields a new infinite family of Heron tetrahedra which correspond to the rational points on a two-parameter elliptic curve.

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“…They proved that tetrahedron with integral edge lengths, rational face areas and rational volume do not exist. If only one face area is forced to be rational then there exist the example (a, b, c, d, e, f ) = (10,8,6,7,11,9), which is conjectured to be unique up to scaling. We have verified this conjecture up to diameter 600000.…”

Section: Perfect Pyramidsmentioning

confidence: 99%

On the generation of Heronian triangles

Kurz¹

2014

Preprint

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We describe several algorithms for the generation of integer Heronian triangles with diameter at most n. Two of them have running time O n 2+ε . We enumerate all integer Heronian triangles for n ≤ 600000 and apply the complete list on some related problems.Lemma 2.1. We can assume that the denominator q can be written as q = w 1 w 2 w 3 w 4 with pairwise coprime integers w 1 , w 2 , w 3 , w 4 and

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Tetrahedra with Integer Edges and Integer Volume

Cited by 5 publications

References 2 publications

Deimination is regulated at multiple levels including auto-deimination of peptidylarginine deiminases

Deimination is regulated at multiple levels including auto-deimination of peptidylarginine deiminases

Rational and Heron tetrahedra

On the generation of Heronian triangles

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