Ostr. Berlin P. 12609
Verena M. Lepper
Artefact metadata
BerlPap-Team; Daniel Werning, Philip Schmitz
TEI encoding
Daniel A. Werning
Sandro Schwarz
Ägyptisches Museum und Papyrussammlung, Berlin
Berlin
v2.1
Licence for this TEI document: Creative Commons, Attribution-ShareAlike 3.0 Unported (CC BY-SA 3.0)
Germany
Berlin
Berlin, Ägyptisches Museum und Papyrussammlung, Staatliche Museen zu Berlin, SPK (P)
Berlin, Ägyptisches Museum und Papyrussammlung, Staatliche Museen zu Berlin, SPK (P)
Papyrussammlung
Ostr. Berlin P. 12609
Ostr. Berlin P. 12609
geometrical exercise: drawing of a regular icosahedron into a sphere of given diameter (cf. Euclid Stoicheia XIII 16)
convex (outside)
Geometrische Aufgabe
clay | fired
ostracon
- object's condition: incomplete
- text's condition:
19 lines, other side uninscribed
Zur Handschrift, vgl. F. A. J. Hoogendijk, Twelve Greek Ostraca from Elephantine, in: F. A. J. Hoogendijk - B. P. Muhs (Hgg.), Sixty-Five Papyrological Texts Presented to Klaas A. Worp on the Occasion of his 65th Birthday, Leiden 2008, 287; zu Z. 3 vgl. H. Cadell, Einleitung zu P.Coll.Youtie I 55–62, S. 337, Anm. 13.
Egypt
Upper Egypt, 1st nome
Elephantine
Rubensohn, Otto (excavation director)
Egypt
Upper Egypt, 1st nome
Elephantine
partage
excavation of O. Rubensohn in Elephantine, winter 1906/07 (26.12.1906)
(unknown)
exhibition | Neues Museum, Berlin
-
1-10, no. 1
209-211
311
253
[http://www.trismegistos.org/tm/detail.php?quick=65672]
[http://smb.museum/berlpap/index.php/16497]
Amulet charm
documentary
account
horizontal format
vertical format
administrative document
bill
commentary
contract
horizontal format
documentary | contract | promissory note
vertical format
dedication
dept instrument
diary
foundation charter
hypomnema
horizontal format
vertical format
indictment
Documentary | Jar Label
legal document
letter
"Urkundenformat"
horizontal format
letter address
private
transversa carta
vertical format
list
name list
onomasticon
product list
marriage contract
names
note
oath
vertical format
offer
documentary | official
documentary | official letter
Order for payment or delivery
documentary | order to arrest
petition to the king (enteuxis, mkmk)
proskynema
act of worhsip wšt.t
name
Documentary | Ration List
receipt
tax receipt
tombstone
will/testament
Documentary | Wooden Label
drawing
commentary
vigniette
literary
biography
commentary
eulogy
historic
lamentation
narrative
prose
verse
poetry
wisdom
instruction
proverb(s)
(no inscription)
(other)
petition to the emperor
Prescription
religious
Christian
commentary
dedication
divination
litany
magical
oracular inquiry
Gnostic
Islamic
religious | Islamic | amulet
dedication
divination
litany
magical
oracular inquiry
ritual
Jewish
dedication
divination
litany
magical
oracular inquiry
ritual
Manichaic
Mesopotamian
polytheistic Aramaian
polytheistic Egyptian
dedication
divination
litany
love charm
magical
oracular inquiry
ritual
unclear religion
dedication
divination
litany
magical
oracular inquiry
religious-literary
Christian
Biblical
New Testament
Old Testament
commentary
eulogy
hymn
lithurgical
monastical
myth
patristic
prayer
Gnostic
commentary
eulogy
hymn
prayer
Islamic
commentary
eulogy
hymn
myth
prayer
Qurʾānic
Jewish
commentary
eulogy
Hebrew Bible
hymn
myth
prayer
Manichaic
commentary
eulogy
hymn
prayer
Mesopotamian
commentary
eulogy
hymn
myth
prayer
polytheistic Aramaian
commentary
eulogy
hymn
myth
prayer
polytheistic Egyptian
Book of the Dead
Coffin Text
commentary
eulogy
hymn
myth
prayer
Pyramid Text
unclear religion
commentary
eulogy
hymn
myth
prayer
scientific
astronomy
Book of Dream
commentary/gloss
math
calculation
essay
geometry
medicine
essay
prescription
scientific | Alchemical presciption
scientific | astrology
horoscop
writing exercise
(unclear)
Akkadian
Arabic
Aramaic, Imperial
Carian
Egyptian
Coptic
Akhmimic
Bohairic
Fayumic
Middle Egyptian / Oxyrhynkhitic
Old Coptic
Sahidic
Sub-Akhmimic
Demotic
Early Demotic
Early Demotic - Ptolemaic Demotic
Ptolemaic - Roman Demotic
Ptolemaic Demotic
Roman Demotic
Late Egyptian
Later Egyptian
Middle Egyptian
Middle Egyptian, Classical
Middle Egyptian, Late
Neo-Middle Egyptian
Old Egyptian
pre-Coptic Egyptian
pre-Coptic Egyptian / Greek
Ptolemaic
Egyptian languages
English
German
Greek, Ancient
Greek, Ancient or Coptic
Hebrew
Hebrew, Ancient
Late Aramaic
Latin
Libyan
Mandaic
Meroitic
Nabataean
(none)
(not identified)
Nubian
Old Nubian
Old Persian
Pahlavi
Phoenician
Semitic language
Syriac
Uninscribed
Late Aramaic
Arabic
Aramaic, Imperial
Carian
Coptic
Coptic Cursive
Coptic Half-uncial
Coptic Uncial
Egyptian
Cursive Hieroglyphs
Demotic
Early Demotic
Early to Middle Demotic
Late Demotic
Middle Demotic
Middle to Late Demotic
Hieratic
Abnormal Hieratic
Archaic Hieratic
Late Hieratic
"Späthieratische Buchschrift"
Middle Hieratic
"Mittelhieratische Buchschrift"
"Mittelhieratische Kanzleischrift"
New Kingdom Hieratic
"Neuhieratische Buchschrift"
"Neuhieratische Kanzleischrift"
Old Hieratic
Hieroglyphs
Cryptography
Hieroglyphs | Late Period
Ptolemaic
Greek
Greek or Coptic
Hebrew
Latin
Meroitic Cursive
Meroitic Hieroglyphs
(none)
(not identified)
Pahlavi
Phoenician
Syriac
uninscribed
Mathematik
Geometrie
Euklid
Elementa
τῆι ηζἰσόπλευρον ἄρα τὸ ζρη. διὰ δὲ ταῦτα καὶ τὰ ησθ
θτι ιυκ κφζ. ἴση δὲ καὶ ἡ ρσ τῆ ι ηθ καὶ ἴσας ἑδείξαμεν
τὰς ρη σθ. ἰσόπλευρον ἄρ' ἑστὶν ἕκαστον τῶν μεταξὺ τριγώνων.
ἴσας γὰρ καὶ παραλλήλους τῆς χλ τὰς ζμ ην θξ ιο κπ ἐγράψαμεν. ἑξα-
γώνου ἄρα καὶ ἡ χρ, ἀλλὰ καὶ ἡ ψχδεκαγώνου, ὥστ' εἶναι τὰς ψρ ψσ
πενταγώνου. ἴση ἄρ' ἡ φρ τῆι ρσ, διὸ
ἰσόπλευρά ἐστι τὰ ψφρ ψρστρίγωνα. διὰ ταῦτα δὲ καὶ ἔκαστον τῶν τὸ ψ ὡς κορυφὴν
ἐχόντων ἰσόπλευρά ἐστιν. πάλιν
ἐκβεβλήσθω
ἐπ' εὐθείας τῆι λψ ἡ λω, ἴση τῆι τοῦ δεκαγώνου πλευρᾶι.
ἤχθω δὴ ἀπὸ τοῦ λ κέντρου ἡ λζ ἑξαγώνου, δεκαγώνου δ' ἦν ἡ
λω, πενταγώνου ἄρ' ἡ ζω, διὰ ταῦτα δ' ἰσόπλευρόν ἐστι
τὰ τρίγωνον τὸ ωζη, διὰ ταὐτὰ
τὸ ζηθικω στερεὸν σχἠμα ἐκ
πρὸς τῶι ω τὰς κορυφὰς ἐχόντων πέντε
τριγώνων ἰσοπλεύρων πεποίηται.
δύο δεκαγώνου πλευραὶ καὶ μία ἑξαγώνου τῆς
δοθείσης σφαίρας
τὴν διάμετρον συμπληροῦσιν. καὶ ἡ τῆς
δοθείσης σφαίρας διάμετρος δυνάμει πενταπλασίων τῆς ἡμι-
διαμέτρου
τοῦ κύκλου ἐστιν τοῦ τὸ πεντάγωνον
περιλαμβανομένου, ἀφ' οὗ τὸ εἰκοσάεδρον ἀναγέγραπται.
ΗΖ ... So the triangle ZΡH is equilateral. Thus also the triangles ΗΣΘ,
ΘΤΙ, ΙΥΚ, ΚΦΖ. But equal is also ΡΣ to the line ΗΘ, and as equal we have shown
the lines ΡΗ and ΣΘ. So each of the triangles lying between (scil. the circles) is equilateral,
because we had drawn the perpendiculars ΖΜ, ΗΝ, ΘΞ, ΙΟ, ΚΠ equal and parallel to the line XΛ.
That is why XP is also (scil. equal to one) hexagonal side. But also ΨΧ was (scil. equal to a) decagonal side, so that ΨΡ and ΨΣ
are pentagonal sides. Equal was <also> ΦΡ to side ΡΣ, therefore ΨΦΡ and ΨΡΣ are equilateral triangles. Therefore each of the triangles whose common apex is Ψ is also
equilateral. Furthermore, let the distance
ΛΩ be drawn as an extension of the distance ΛΨ, equal to the side of the decagon.
Let the radius ΛΖ equal to the hexagon side be drawn from the centre of the circle Λ, but (scil. equal) to the decagon side was
ΛΩ , so ΖΩ is a pentagon side. Therefore equilateral is
the triangle ΩΖΗ. For the same reason, the spatial figure ΖΗΘΙΚΩ is formed by
five equilateral triangles having their apex in Ω.
Two decagonal sides and one hexagonal side together
give the diameter of the given sphere cf. Euclid El. XIII 16 corollary,
and the diameter of the given sphere squared is five times
the radius of the given circle squared,
drawn around the pentagon on which the icosahedron was built. (after Müller/Mau 1960)