CDL Specification

Complete reference for the Crystal Description Language (CDL) syntax and semantics.

Syntax Overview

CDL uses a compact notation to describe crystal morphology:

system[point_group]:{form}@scale + {form}@scale | modification | twin(law)

Components

Component	Required	Description
system	Yes	Crystal system name
[point_group]	No	Point group (defaults to highest symmetry)
:{form}	Yes	At least one crystal form
@scale	No	Scale factor (default: 1.0)
+ {form}	No	Additional forms
\| modification	No	Shape modifications
\| twin(law)	No	Twin specification

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Crystal Systems

All 7 crystal systems are supported:

System	Axes	Angles	Default Point Group
cubic	a = b = c	α = β = γ = 90°	m3m
tetragonal	a = b ≠ c	α = β = γ = 90°	4/mmm
orthorhombic	a ≠ b ≠ c	α = β = γ = 90°	mmm
hexagonal	a = b ≠ c	α = β = 90°, γ = 120°	6/mmm
trigonal	a = b ≠ c	α = β = 90°, γ = 120°	-3m
monoclinic	a ≠ b ≠ c	α = γ = 90°, β ≠ 90°	2/m
triclinic	a ≠ b ≠ c	α ≠ β ≠ γ ≠ 90°	-1

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Point Groups

Complete Reference

Cubic System (5 point groups)

Point Group	Symmetry	Operations	Example Minerals
m3m	Full cubic	48	Diamond, garnet, fluorite
432	Rotations only	24	Sal ammoniac
-43m	Tetrahedral	24	Sphalerite, tetrahedrite
m-3	With center	24	Pyrite
23	Minimal cubic	12	Ullmannite

Tetragonal System (7 point groups)

Point Group	Symmetry	Operations	Example Minerals
4/mmm	Full tetragonal	16	Zircon, rutile
422	Rotations	8	Phosgenite
4mm	Pyramidal	8	Diabolite
-42m	Scalenohedral	8	Chalcopyrite
4/m	With center	8	Scheelite
-4	Tetragonal disphenoidal	4	Cahnite
4	Pyramidal	4	Wulfenite

Hexagonal System (7 point groups)

Point Group	Symmetry	Operations	Example Minerals
6/mmm	Full hexagonal	24	Beryl, apatite
622	Trapezoidal	12	High quartz
6mm	Pyramidal	12	Wurtzite
-6m2	Ditrigonal dipyramidal	12	Benitoite
6/m	Dipyramidal	12	Apatite
-6	Trigonal dipyramidal	6	-
6	Pyramidal	6	Nepheline

Trigonal System (5 point groups)

Point Group	Symmetry	Operations	Example Minerals
-3m	Ditrigonal scalenohedral	12	Calcite, corundum
32	Trapezoidal	6	Quartz, cinnabar
3m	Ditrigonal pyramidal	6	Tourmaline
-3	Rhombohedral	6	Dolomite
3	Trigonal pyramidal	3	-

Orthorhombic System (3 point groups)

Point Group	Symmetry	Operations	Example Minerals
mmm	Full orthorhombic	8	Topaz, olivine
222	Disphenoidal	4	Epsomite
mm2	Pyramidal	4	Hemimorphite

Monoclinic System (3 point groups)

Point Group	Symmetry	Operations	Example Minerals
2/m	Prismatic	4	Orthoclase, gypsum
m	Domatic	2	Clinohedrite
2	Sphenoidal	2	Sucrose

Triclinic System (2 point groups)

Point Group	Symmetry	Operations	Example Minerals
-1	Pinacoidal	2	Albite, turquoise
1	Pedial	1	-

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Miller Indices

3-Index Notation (hkl)

Standard notation for cubic, tetragonal, orthorhombic, monoclinic, and triclinic systems:

{hkl}     # General form
{111}     # Octahedron in cubic
{100}     # Cube in cubic
{110}     # Dodecahedron in cubic
{211}     # Trapezohedron in cubic

4-Index Notation (hkil) - Miller-Bravais

Used for hexagonal and trigonal systems where i = -(h+k):

{hkil}      # General form (i is redundant but conventional)
{10-10}     # Hexagonal prism (h=1, k=0, i=-1, l=0)
{0001}      # Basal pinacoid
{10-11}     # Rhombohedron
{11-20}     # Second-order prism

Notation Rules

Format	Example	Description
Condensed	{111}	Single digits concatenated
Spaced	{1 1 1}	Space-separated (multi-digit)
Negative	{10-11}	Minus before digit
Multi-digit	{12 3 4}	Space required

Common Cubic Forms

Miller Index	Name	Faces
{111}	Octahedron	8
{100}	Cube	6
{110}	Dodecahedron	12
{211}	Trapezohedron	24
{221}	Trisoctahedron	24
{311}	Tetrahexahedron	24
{210}	Tetrahexahedron	24
{321}	Hexoctahedron	48

Common Hexagonal/Trigonal Forms

Miller Index	Name	Faces
{0001}	Basal pinacoid	2
{10-10}	First-order prism	6
{11-20}	Second-order prism	6
{10-11}	Positive rhombohedron	6
{01-11}	Negative rhombohedron	6
{10-12}	Positive scalenohedron	12
{11-21}	Trigonal dipyramid	6

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Named Forms

Named forms can be used instead of Miller indices:

Cubic Named Forms

Name	Miller Index	Description
octahedron	{111}	8-faced regular solid
cube	{100}	6-faced regular solid
dodecahedron	{110}	12-faced rhombic solid
trapezohedron	{211}	24-faced solid
trisoctahedron	{221}	24 triangular faces
tetrahexahedron	{210}	24 quadrilateral faces
hexoctahedron	{321}	48-faced solid

Hexagonal/Trigonal Named Forms

Name	Miller Index	Description
prism	{10-10}	Hexagonal prism
prism_1	{10-10}	First-order prism
prism_2	{11-20}	Second-order prism
basal	{0001}	Basal pinacoid
pinacoid	{0001}	Same as basal
rhombohedron	{10-11}	6-faced form
rhombohedron_r	{10-11}	Positive rhombohedron
rhombohedron_z	{01-11}	Negative rhombohedron
pyramid	{10-11}	Hexagonal pyramid
dipyramid	{10-11}	Hexagonal dipyramid
scalenohedron	{21-31}	12-faced form

Tetragonal Named Forms

Name	Miller Index	Description
prism	{100}	Tetragonal prism
prism_1	{100}	First-order prism
prism_2	{110}	Second-order prism
pyramid	{101}	Tetragonal pyramid
dipyramid	{101}	Tetragonal dipyramid
bipyramid	{111}	Tetragonal bipyramid

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Scale Values

The @scale parameter controls form prominence:

{111}@1.0 + {100}@1.3   # Cube truncates octahedron
{111}@1.0 + {100}@0.3   # Small cube facets on octahedron

Interpretation

• Larger scale = Face is further from center = More prominent
• Smaller scale = Face is closer to center = Less prominent
• Equal scales = Forms meet at equal distances

Recommended Ranges

Purpose	Scale Range
Dominant form	1.0
Moderate truncation	0.3 - 0.8
Heavy truncation	1.2 - 2.0
Subtle modification	0.1 - 0.3

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Modifications

Modifications alter the crystal shape after form generation.

elongate

Stretch the crystal along an axis:

{111} | elongate(c:1.5)    # Stretch 1.5x along c-axis
{111} | elongate(a:2.0)    # Stretch 2x along a-axis
{111} | elongate(-c:0.8)   # Compress along negative c

Parameters:

Parameter	Values	Description
axis	a, b, c, +a, -a, etc.	Axis direction
ratio	0.1 - 10.0	Stretch factor

truncate

Truncate edges or vertices:

{111} | truncate(vertex:0.2)   # Truncate vertices
{111} | truncate(edge:0.3)     # Truncate edges
{111} | truncate({100}:0.5)    # Truncate with specific form

Parameters:

Parameter	Values	Description
target	vertex, edge, {hkl}	What to truncate
depth	0.0 - 1.0	Truncation depth

taper

Create a tapered crystal shape:

{111} | taper(+c:0.8)    # Taper toward +c direction
{111} | taper(-c:0.5)    # Taper toward -c direction

Parameters:

Parameter	Values	Description
direction	+a, -a, +b, -b, +c, -c	Taper direction
factor	0.0 - 1.0	Taper amount (0 = no taper, 1 = point)

bevel

Bevel edges of the crystal:

{111} | bevel(all:0.1)     # Bevel all edges
{111} | bevel(top:0.2)     # Bevel top edges only

Parameters:

Parameter	Values	Description
edges	all, top, bottom, vertical	Which edges
width	0.0 - 0.5	Bevel width

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Twin Specifications

Twins are described using the twin() modification.

Contact Twins

Two crystals sharing a composition plane:

{111} | twin(spinel)       # Spinel law twin
{10-10} | twin(japan)      # Japan law quartz twin
{10-10} | twin(brazil)     # Brazil law quartz twin

Penetration Twins

Two crystals interpenetrating:

{100} | twin(fluorite)     # Fluorite interpenetration
{210} | twin(iron_cross)   # Pyrite iron cross

Cyclic Twins

Multiple crystals in cyclic arrangement:

{10-11} | twin(trilling, 3)   # Three-fold cyclic twin
{10-11} | twin(sixling, 6)    # Six-fold cyclic twin

Complete Twin Laws Reference

Law	Type	System	Description
spinel	Contact	Cubic	Twin on {111}, common in spinel
spinel_law	Contact	Cubic	Same as spinel
iron_cross	Penetration	Cubic	{110} twin, pyrite
fluorite	Penetration	Cubic	Interpenetration twin
brazil	Contact	Trigonal	Optical twin in quartz
dauphine	Contact	Trigonal	Electrical twin in quartz
japan	Contact	Trigonal	{11-22} twin in quartz
carlsbad	Penetration	Monoclinic	Common in orthoclase
baveno	Contact	Monoclinic	{021} twin in feldspar
manebach	Contact	Monoclinic	{001} twin in feldspar
albite	Contact	Triclinic	Polysynthetic in plagioclase
pericline	Contact	Triclinic	Twin in albite
gypsum_swallow	Penetration	Monoclinic	Swallow-tail in gypsum
staurolite_60	Penetration	Monoclinic	60° cross
staurolite_90	Penetration	Monoclinic	90° cross
trilling	Cyclic	Any	Three individuals

Custom Twin Axis

Specify custom twin axis and angle:

{111} | twin([1,1,1], 180, contact)    # Custom axis
{111} | twin([1,0,0], 90, penetration) # 90° rotation about a-axis

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Complete Examples

Diamond

cubic[m3m]:{111}@1.0 + {110}@0.2

Octahedron with small dodecahedron modification.

Garnet

cubic[m3m]:{110}@1.0 + {211}@0.6

Dodecahedron with trapezohedron.

Quartz

trigonal[32]:{10-10}@1.0 + {10-11}@0.8 + {01-11}@0.8

Prism with positive and negative rhombohedra.

Ruby/Sapphire

trigonal[-3m]:{10-10}@1.0 + {0001}@0.3 + {10-11}@0.5

Hexagonal prism with pinacoid and rhombohedron.

Beryl/Emerald

hexagonal[6/mmm]:{10-10}@1.0 + {0001}@0.5

Hexagonal prism with basal pinacoid.

Spinel Twin

cubic[m3m]:{111} | twin(spinel)

Twinned octahedron (macle).

Japan Law Quartz Twin

trigonal[32]:{10-10}@1.0 + {10-11}@0.8 | twin(japan)

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Grammar (EBNF)

cdl         = system, [point_group], ":", forms, {modification} ;
system      = "cubic" | "tetragonal" | "orthorhombic"
            | "hexagonal" | "trigonal" | "monoclinic" | "triclinic" ;
point_group = "[", symbol, "]" ;
forms       = form, {"+" , form} ;
form        = (miller | named_form), ["@", scale] ;
miller      = "{", indices, "}" ;
indices     = index, index, index, [index] ;  (* 3 or 4 indices *)
index       = ["-"], digit, {digit} ;
named_form  = letter, {letter | "_" | digit} ;
scale       = number ;
modification = "|", (mod_call | twin_call) ;
mod_call    = mod_name, "(", parameters, ")" ;
twin_call   = "twin", "(", twin_spec, ")" ;
twin_spec   = twin_law, [",", count] | custom_twin ;
custom_twin = "[", axis, "]", ",", angle, ",", twin_type ;

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Error Messages

Parse Errors

Error	Cause	Solution
"Unknown crystal system"	Invalid system name	Check spelling
"Invalid point group for system"	Mismatch	Use correct point group
"Invalid Miller index"	Bad notation	Check index format
"Expected ':'"	Missing colon	Add colon after system
"Unexpected character"	Syntax error	Check special characters

Validation Errors

Error	Cause	Solution
"4-index constraint violated"	i ≠ -(h+k)	Fix Miller-Bravais index
"Unknown twin law"	Invalid twin name	Check twin law spelling
"Unknown modification"	Invalid mod name	Use valid modification