Geometrical Patterns in the Pre-classical Greek Area. Prospecting the Borderland between Decoration, Art, and Structural Inquiry

Jens Høyrup

Revue d'histoire des mathématiques (2000)

  • Volume: 6, Issue: 1, page 1-54
  • ISSN: 1262-022X

Abstract

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Many general histories of mathematics mention prehistoric “geometric” decorations along with counting and tally-sticks as the earliest beginnings of mathematics, insinuating thus (without making it too explicit) that a direct line of development links such decorations to mathematical geometry. The article confronts this persuasion with a particular historical case: the changing character of geometrical decorations in the later Greek area from the Middle Neolithic through the first millennium bce.The development during the “Old European” period (sixth through third millennium bce, calibrated radiocarbon dates) goes from unsystematic and undiversified beginnings toward great phantasy and variation, and occasional suggestions of combined symmetries, but until the end largely restricted to the visually prominent, and not submitted to formal constraints; the type may be termed “geometrical impressionism”.Since the late sixth millennium, spirals and meanders had been important. In the Cycladic and Minoan orbit these elements develop into seaweed and other soft, living forms. The patterns are vitalized and symmetries dissolve. One might speak of a change from decoration into art which, at the same time, is a step away from mathematical geometry.Mycenaean Greece borrows much of the ceramic style of the Minoans; other types of decoration, in contrast, display strong interest precisely in the formal properties of patterns — enough, perhaps, to allow us to speak about an authentically mathematical interest in geometry. In the longer run, this has a certain impact on the style of vase decoration, which becomes more rigid and starts containing non-figurative elements, without becoming really formal. At the breakdown of the Mycenaean state system around 1200 bce, the “mathematical” formalization disappears, and leaves no trace in the decorations of the subsequent Geometric period. These are, instead, further developments of the non-figurative elements and the repetitive style of late Mycenaean vase decorations. Instead of carrying over mathematical exploration from the early Mycenaean to the Classical age, they represent a gradual sliding-back into the visual geometry of earlier ages.The development of geometrical decoration in the Greek space from the Neolithic through the Iron Age is thus clearly structured when looked at with regard to geometric conceptualizations and ideals. But it is not linear, and no necessity leads from geometrical decoration toward geometrical exploration of formal structures (whether intuitive or provided with proofs). Classical Greek geometry, in particular, appears to have its roots much less directly (if at all) in early geometrical ornamentation than intimated by the general histories.

How to cite

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Høyrup, Jens. "Geometrical Patterns in the Pre-classical Greek Area. Prospecting the Borderland between Decoration, Art, and Structural Inquiry." Revue d'histoire des mathématiques 6.1 (2000): 1-54. <http://eudml.org/doc/252056>.

@article{Høyrup2000,
abstract = {Many general histories of mathematics mention prehistoric “geometric” decorations along with counting and tally-sticks as the earliest beginnings of mathematics, insinuating thus (without making it too explicit) that a direct line of development links such decorations to mathematical geometry. The article confronts this persuasion with a particular historical case: the changing character of geometrical decorations in the later Greek area from the Middle Neolithic through the first millennium bce.The development during the “Old European” period (sixth through third millennium bce, calibrated radiocarbon dates) goes from unsystematic and undiversified beginnings toward great phantasy and variation, and occasional suggestions of combined symmetries, but until the end largely restricted to the visually prominent, and not submitted to formal constraints; the type may be termed “geometrical impressionism”.Since the late sixth millennium, spirals and meanders had been important. In the Cycladic and Minoan orbit these elements develop into seaweed and other soft, living forms. The patterns are vitalized and symmetries dissolve. One might speak of a change from decoration into art which, at the same time, is a step away from mathematical geometry.Mycenaean Greece borrows much of the ceramic style of the Minoans; other types of decoration, in contrast, display strong interest precisely in the formal properties of patterns — enough, perhaps, to allow us to speak about an authentically mathematical interest in geometry. In the longer run, this has a certain impact on the style of vase decoration, which becomes more rigid and starts containing non-figurative elements, without becoming really formal. At the breakdown of the Mycenaean state system around 1200 bce, the “mathematical” formalization disappears, and leaves no trace in the decorations of the subsequent Geometric period. These are, instead, further developments of the non-figurative elements and the repetitive style of late Mycenaean vase decorations. Instead of carrying over mathematical exploration from the early Mycenaean to the Classical age, they represent a gradual sliding-back into the visual geometry of earlier ages.The development of geometrical decoration in the Greek space from the Neolithic through the Iron Age is thus clearly structured when looked at with regard to geometric conceptualizations and ideals. But it is not linear, and no necessity leads from geometrical decoration toward geometrical exploration of formal structures (whether intuitive or provided with proofs). Classical Greek geometry, in particular, appears to have its roots much less directly (if at all) in early geometrical ornamentation than intimated by the general histories.},
author = {Høyrup, Jens},
journal = {Revue d'histoire des mathématiques},
keywords = {beginnings of mathematics; pre-classical Greek artefacts; protogeometric art; sources of theoretical geometry},
language = {eng},
number = {1},
pages = {1-54},
publisher = {Société mathématique de France},
title = {Geometrical Patterns in the Pre-classical Greek Area. Prospecting the Borderland between Decoration, Art, and Structural Inquiry},
url = {http://eudml.org/doc/252056},
volume = {6},
year = {2000},
}

TY - JOUR
AU - Høyrup, Jens
TI - Geometrical Patterns in the Pre-classical Greek Area. Prospecting the Borderland between Decoration, Art, and Structural Inquiry
JO - Revue d'histoire des mathématiques
PY - 2000
PB - Société mathématique de France
VL - 6
IS - 1
SP - 1
EP - 54
AB - Many general histories of mathematics mention prehistoric “geometric” decorations along with counting and tally-sticks as the earliest beginnings of mathematics, insinuating thus (without making it too explicit) that a direct line of development links such decorations to mathematical geometry. The article confronts this persuasion with a particular historical case: the changing character of geometrical decorations in the later Greek area from the Middle Neolithic through the first millennium bce.The development during the “Old European” period (sixth through third millennium bce, calibrated radiocarbon dates) goes from unsystematic and undiversified beginnings toward great phantasy and variation, and occasional suggestions of combined symmetries, but until the end largely restricted to the visually prominent, and not submitted to formal constraints; the type may be termed “geometrical impressionism”.Since the late sixth millennium, spirals and meanders had been important. In the Cycladic and Minoan orbit these elements develop into seaweed and other soft, living forms. The patterns are vitalized and symmetries dissolve. One might speak of a change from decoration into art which, at the same time, is a step away from mathematical geometry.Mycenaean Greece borrows much of the ceramic style of the Minoans; other types of decoration, in contrast, display strong interest precisely in the formal properties of patterns — enough, perhaps, to allow us to speak about an authentically mathematical interest in geometry. In the longer run, this has a certain impact on the style of vase decoration, which becomes more rigid and starts containing non-figurative elements, without becoming really formal. At the breakdown of the Mycenaean state system around 1200 bce, the “mathematical” formalization disappears, and leaves no trace in the decorations of the subsequent Geometric period. These are, instead, further developments of the non-figurative elements and the repetitive style of late Mycenaean vase decorations. Instead of carrying over mathematical exploration from the early Mycenaean to the Classical age, they represent a gradual sliding-back into the visual geometry of earlier ages.The development of geometrical decoration in the Greek space from the Neolithic through the Iron Age is thus clearly structured when looked at with regard to geometric conceptualizations and ideals. But it is not linear, and no necessity leads from geometrical decoration toward geometrical exploration of formal structures (whether intuitive or provided with proofs). Classical Greek geometry, in particular, appears to have its roots much less directly (if at all) in early geometrical ornamentation than intimated by the general histories.
LA - eng
KW - beginnings of mathematics; pre-classical Greek artefacts; protogeometric art; sources of theoretical geometry
UR - http://eudml.org/doc/252056
ER -

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