Symmetry
Vitruvian symmetry
Concept: Symmetry as the appropriate harmony arising from the details of a work, where each detail corresponds to the whole design (eurhythmy).
Historical anchors: Vitruvius (1st c. B.C.E., De Architectura, Book III) and architectural examples such as the Parthenon (c. 447–432 B.C.E.) and the Pantheon (c. 125 B.C.E.).
Basic idea: Architecture should mirror the human body; symmetry emerges from the correspondence between parts and the whole form.
Vitruvius quote (paraphrased): The measures of a man are distributed by nature so that 4 digits make 1 palm, 4 palms make 1 foot, 6 palms make a cubit, 4 cubits make 1 man, 4 cubits make 1 step, and 24 palms make 1 man; these measures appear in constructions.
Leonardo da Vinci’s Vitruvian Man (c. 1490): inspired by Vitruvius’ De Architectura; the navel is at the center; if a circle is described around the navel with the man lying flat and limbs extended, the circle touches fingers and toes; the same figure fits within a square by measuring feet-to-crown and arm span; lines at right angles enclose the figure into a square.
Leonardo’s mirror writing: Original Italian and its English translation emphasize Vitruvius’ claim about human body measures distributed by nature.
Key takeaway: Vitruvian symmetry relates proportion, geometry, and human anatomy to architectural form; eurhythmy between detail and whole drives design.
Anthropometric and Vitruvian proportions
Proportions of the ideal human body (anthropometric system) used to compare Imperial vs Metric measures:
3 inches ≈ 7.5 cm
12 inches ≈ 30 cm
18 inches ≈ 45 cm
72 inches ≈ 1.8 m
Note: A table attempting to align Imperial and Metric measures is partially garbled in the transcript; the above values illustrate the general idea of body-part scaling used in Vitruvian proportion studies.
Takeaway: The Vitruvian ideal uses standardized body proportions as a geometric basis for design, reinforcing the link between human form and architectural symmetry.
Geometrical symmetry: Bilaterians and basic animal plan
Symmetry in geometry: Invariance under geometrical transformations.
Biological fact: A human being is a mammal; all mammals are bilaterians (have a through-gut with a distinct mouth and anus).
Bilaterian classification:
Protostomes: mouth forms first (Greek proto- = first, stoma = mouth).
Deuterostomes: anus forms first (Greek deutero- = second).
Common bilaterian ancestor (per Arendt et al., 2016).
Bilaterians share a body plan with bilateral symmetry (left-right orientation) and a through-gut, which underpins their early developmental symmetry.
Examples listed (as typical protostomes/deuterostomes and representatives):
Insects, Molluscs, Flatworms (examples of protostomes or early bilaterians)
Mammals (examples of deuterostomes)
Significance: Bilateral symmetry is a fundamental geometric property linked to development, locomotion, and body organization across diverse animal lineages.
Geometrical symmetry: Reflectional symmetry
Definition: Invariance under reflection along a line of symmetry.
Concept: An object with reflectional symmetry maps onto itself when flipped across a line.
Vitruvian Man: A primary example of partial reflectional symmetry (bilateral symmetry) in 2D.
Regular polygons: Have n lines of symmetry where n is the number of sides:
Equilateral triangle: n = 3
Square: n = 4
Regular pentagon: n = 5
Regular hexagon: n = 6
Geometrical symmetry: Circles and the limit of polygons
As n → ∞, a regular n-sided polygon becomes indistinguishable from a circle.
A circle has an infinite number of lines of symmetry in 2D; thus, it has full reflectional symmetry along any line through its center.
The circle in the Vitruvian display is more symmetrical than the Vitruvian Man itself when considered as a geometric background.
Conclusion: Circles are the ultimate symbol of maximal reflectional symmetry in 2D.
Geometrical symmetry: Rotational (radial) symmetry
Definition: Invariance under rotation about the center of the figure.
Relation to reflectional symmetry: Full reflectional symmetry implies full rotational symmetry in 2D (and vice versa under certain conditions).
Examples by degree of rotation:
3-fold rotational symmetry: Mercedes-Benz logo (rotates by 120°)
4-fold rotational symmetry: Windows 11 logo (rotates by 90°)
6-fold rotational symmetry: Snowflake (rotates by 60° increments)
Circle: full rotational symmetry for all angles (360° all around)
Geometrical symmetry: Circles, spheres, and translational symmetry
Circle: Infinite rotational and reflectional symmetry in 2D; no translational symmetry.
Sphere: Full reflectional symmetry along x-, y-, and z-axes in 3D; more symmetric than a circle in 3D terms.
Translational symmetry: Invariance under spatial translation; neither circle nor sphere possess translational symmetry on their own.
Tessellations (tiled patterns): Have translational symmetry; examples include rhombille tilings and Escher’s patterns.
Important caveat: Isolated shapes do not have translational symmetry.
Escher example: Regular Division of The Plane with Birds (1949, woodcut).
Geometrical symmetry: Full rotational and translational structure in patterns
Full rotational symmetry and translational symmetry cannot generally coexist in simple shapes like circles (in 2D) due to tessellation constraints.
Crystallographic restriction theorem: Crystals modeled as lattice patterns have rotational symmetries limited to 2-, 3-, 4-, or 6-fold.
Penrose tiling (aperiodic): Uses rhombi to exhibit 5-fold rotational symmetry without translational symmetry.
Honeycomb tiling (periodic): Exhibits 6-fold rotational symmetry with translational symmetry.
Why 5-fold symmetry is forbidden for crystals
Reason: Crystals repeat periodically in a lattice.
Geometric proof sketch (pentagon): Interior angle of a regular pentagon = heta = rac{(5-2)}{5} imes 180^ op = 108^ op
To tile around a vertex, the sum of angles must be an integer multiple of 360°; here 360/108 ≈ 3.333, which is not an integer.
Therefore, regular pentagons cannot tile the plane in a connected lattice.
In macroscopic life, structures are not built by repeating crystal lattices, so 5-fold symmetry appears in organisms (e.g., starfish, flowers).
Conclusion: 5-fold rotational symmetry is forbidden for crystals, but it occurs in biology and art.
Vitruvian man, circle, and tiling: a quick synthesis
Vitruvian man illustrates partial reflectional symmetry (2D).
A circle exhibits full reflectional and rotational symmetry (2D).
Honeycomb tiling demonstrates 6-fold rotational symmetry with translational symmetry (periodic).
Pure flat Euclidean space exhibits full reflectional and rotational symmetry with translational symmetry.
In contrast, pure Minkowski (empty spacetime) space (before universe began) is “featureless” and maximally symmetrical but has no structure.
Physical symmetry: Gravity, relativity, and spacetime
Gravity is extremely weak relative to electromagnetic, weak, and strong forces; thus gravitational effects are negligible in everyday life unless masses are large (e.g., planets, stars, galaxies, black holes).
Special relativity (1905): Laws of physics take the same form in all inertial frames of reference; the speed of light in vacuum is constant for all observers, c \approx 3 \times 10^8 \,\mathrm{m/s}.
General relativity (1915): Gravity is geometry of spacetime; spacetime is curved by mass-energy.
Einstein's field equations (schematic): G{\mu\nu} + \Lambda g{\mu\nu} = \frac{8\pi G}{c^4} T{\mu\nu} where G{\mu\nu} describes spacetime curvature, g{\mu\nu} is the metric, and T{\mu\nu} is the stress-energy tensor.
Einstein’s “first postulate” and the equivalence principle underlie local inertial frames; curvature encodes gravitational effects.
Planetary orbits are geodesics in curved spacetime; spacetime tells matter how to move, and matter tells spacetime how to curve (John Wheeler, 1973).
Spacetime symmetries: Global vs local
In special relativity (Minkowski spacetime), global symmetry under Poincaré group applies everywhere.
In general relativity, spacetime is curved and dynamic; global symmetry generally does not apply; instead, there is local symmetry that can vary from point to point.
If there is no matter or energy, spacetime is flat (special relativity case).
Key idea: Global symmetry breaks down in curved spacetime; local symmetry remains central in GR.
The Minkowski picture and the Poincaré group
Description: Special relativity can be described in Minkowski geometry with coordinates of an event given by (ct, x, y, z) .
In truly empty space, Minkowski space exhibits full Poincaré symmetry—the 10-parameter group of spacetime isometries.
The Poincaré symmetries include:
3 spatial translations: along x, y, z
1 time translation: along the temporal axis
3 spatial rotations: roll, pitch, yaw
3 Lorentz boosts: along x, y, z
Total number of Poincaré symmetries: 3 + 1 + 3 + 3 = 10
Noether’s theorem: continuous symmetries and conserved quantities
Core claim (Noether, 1918): Every continuous symmetry in nature corresponds to a conservation law.
Illustrative correspondences:
Rotational symmetry ⇄ Angular momentum: \mathbf{L} = I \, \boldsymbol{\omega} where I is the moment of inertia and \boldsymbol{\omega} is angular velocity.
Translational symmetry in space ⇄ Linear momentum: \mathbf{p} = m \mathbf{v}; Conservation of momentum.
Translational symmetry in time ⇄ Energy: E_{\text{total}} = K + U; Conservation of energy.
Examples:
Figure skater pulling arms in: moment of inertia decreases (I ↓) and angular velocity increases (ω ↑) to conserve angular momentum when arms are drawn inward.
Cue ball and target ball in pool: total momentum before and after collision is conserved, i.e., \mathbf{p}{\text{total}} = \mathbf{p}1 + \mathbf{p}_2 remains constant.
Summary: Noether’s theorem links continuous spacetime symmetries to conserved physical quantities, revealing a deep structural unity in physics.
Epigraph: Emmy Noether celebrated for foundational contributions; Einstein’s obituary quotes praise for her mathematical genius and the unifying power of her ideas in algebra and physics.
Noether’s theorem applied to the Poincaré group and energy-momentum
When applied to the Poincaré group, Noether’s theorem implies the conservation of energy, momentum, and angular momentum as fundamental invariants of spacetime and matter.
Historical note: Noether’s work is celebrated for revealing a deep mathematical structure underpinning conservation laws; Einstein’s obituary (1935) praises her contributions to mathematics and physics.
Symmetry re-considered: dissipative structures and complexity
Although Minkowski space is highly symmetric, real systems are open and exchange matter/energy with their environment.
Prigogine’s theory of dissipative structures (with Lefever, Nicolis, Stengers) describes self-organization in open systems that maintain order by exchanging matter or energy with their surroundings.
These complex, dissipative structures arise from symmetry-breaking and include:
Artistic creations
Human-made structures
Living organisms
Crystals
The structure of galaxies
Empty spacetime (before structure emerged)
Takeaway: Symmetry breaking leads to new levels of organization and complexity; symmetry is a guiding principle, but real-world systems evolve by breaking and reorganizing that symmetry.
Symmetry in molecules and chemical complexity
Ball-and-stick models and molecular formulas (examples):
Ammonia: NH3; number of atoms = 4; molar mass ≈ 17.031 g/mol
Hydrogen phosphide (phosphine): PH3; number of atoms = 4; molar mass ≈ 34.0 g/mol
Sucrose: C12H22O11; number of atoms = 45; molar mass ≈ 342.3 g/mol
Color-coding and molecular complexity (CPK coloring) illustrate symmetry in molecules and how arrangement of atoms affects properties.
Note: The transcript includes a molecular symmetry discussion and the idea that larger, more complex molecules have richer symmetry-related properties.
Symmetry and complexity: pyramids, inversion, and chirality
Ammonia (NH3) inversion: The molecule can invert between two pyramidal configurations (Pyramid 1 and Pyramid 2).
Nitrogen atom can tunnel through the plane of hydrogen atoms (quantum tunneling) at a rate higher than classical predictions, causing rapid inversion between two equivalent pyramidal forms.
Comparison with phosphine (PH3): Heavier phosphorus makes inversion slower; frequency is about 1/10 of NH3’s rate, so inversion is much less rapid.
Sucrose and other sugars: Chiral molecules are not superimposable on their mirror images; D- and L- forms exist.
Chirality and homochirality in life
Definition: A system is chiral if it is distinguishable from its mirror image and cannot be superimposed onto it (e.g., human hands).
D- and L- nomenclature: D for dexter (right), L for laevus (left).
Sugar and amino acid chirality:
Sucrose is chiral; its mirror image (fructose/glucose isomers) exists with different structures.
D-sucrose vs L-sucrose: Only D-sucrose occurs naturally; L-sucrose is not common in nature.
Homochirality of life: Life uses only one hand of chiral forms for essential biomolecules:
Sugars: predominantly D-form
Amino acids: predominantly L-form
Parity symmetry (invariance under spatial inversion) is broken in biology to produce a single handedness, enabling the consistent assembly of macromolecules.
There are 9 possible amino acid–sugar configuration patterns; life’s preference for a single-handed set underpins biochemistry.
Hierarchy of sciences and physicalism
Anderson’s hierarchy of sciences (1972) argues for a cascade of levels of complexity:
Solid-state or many-body physics → Elementary particle physics → Chemistry → Molecular biology → Cell biology → Molecular biology → Psychology → Physiology → Social sciences → Psychology (increasing complexity)
The idea is that higher-level sciences exhibit emergent properties not reducible to lower-level laws, even though naming and understanding at lower levels informs higher ones.
Physicalism and reductionism debates (32–33):
Rutherford-like claim: “In science, there is only physics; all the rest is stamp collecting.” (historical attributed view on physical reductionism)
Descartes (1641) and Berkeley (1710) presented non-physicalist and idealist positions:
Descartes: mind-body dualism (non-physical mind cannot be reduced to physics)
Berkeley: idealism (everything reduces to ideas in the mind of God)
Fodor (1974) argued for non-reductionist yet physically grounded views of the special sciences; the autonomy of higher sciences without complete reduction to physics.
Wilson (1988) advocates for a unified, integrative science; nonetheless, he supports reductionism at higher levels in principle, while acknowledging complexity.
Takeaway: Debates about how much of science can/should be reduced to physics reflect deeper questions about explanation, autonomy of disciplines, and emergent phenomena.
Appendix: Python code for generating n-sided polygons
Purpose: Generate and plot regular n-gons for a range of n (3 to 20) to visualize geometric symmetry.
High-level description of the code (summarized):
Uses matplotlib and numpy to create a grid of subplots.
draw_polygon(ax, n): computes vertices on a unit circle at angles evenly spaced by \theta = \frac{2\pi k}{n} for k = 0,…,n-1, closes the polygon by appending the first vertex, and fills it with color.
Loops over n from 3 to 20, calling draw_polygon for each subplot.
Ensures equal aspect ratio and hides axes for a clean visualization.
The code demonstrates practical visualization of rotational and reflectional symmetry in regular polygons.
Notable quotes and concepts mentioned
Vitruvius on proportion and symmetry: the human body as the model for architectural design; eurhythmy as harmony between parts and whole.
John Wheeler (1973): “Spacetime tells matter how to move; matter tells spacetime how to curve.”
Einstein (1935) obituary for Emmy Noether: praise for her mathematical genius and the unity she revealed in symmetry and conservation laws.
Prigogine and dissipative structures: self-organization through symmetry-breaking in open systems, yielding complex structures including life, art, and galaxies.
Quick reference: key formulas and constants
Circle and pentagon geometry:
Regular pentagon interior angle: \theta = \frac{(5-2)}{5} \times 180^ op = 108^ op
Tiling constraint around a vertex: 360^ op / \theta \approx 3.333 \notin \mathbb{Z}
Poincaré group (symmetries in spacetime): total of 3 + 1 + 3 + 3 = 10 symmetry operations.
Noether’s theorem correspondences:
Angular momentum: \mathbf{L} = I \, \boldsymbol{\omega}
Linear momentum: \mathbf{p} = m \, \mathbf{v}
Energy: E_{\text{total}} = K + U
Einstein field equation (schematic): G{\mu\nu} + \Lambda g{\mu\nu} = \frac{8\pi G}{c^4} \, T_{\mu\nu}
Minkowski coordinates for SR: (ct, x, y, z)
Light speed: c \approx 3 \times 10^8 \, \text{m/s}