Platonic Solids ── A Puzzle To The Chemists: Exploring ThroughSymmetry, Group Symmetry and Group Theory

Priyatosh DuttaAnanda Mohan College

ABSTRACTSymmetry is almost as ubiquitous and mystic as the concept of GOD (!), is as old as the time eternity. It is nearly everywhere, atomic world through the universe, is manifested in almost everything, through either geometric symmetry (forms, structure, etc.) or through physical symmetry (natural laws, audible symmetry in music etc.). It plays its role at fundamental level in all human creative endeavour including Science, Art, Music and what not! Since ancient times, people have identified the notion of symmetry with a wide spectrum of ideas or concepts, starting from equity, balance through beauty, truth, etc. until, only recently, they reached a concrete, mathematical definition that made symmetry something for purpose too. Consequently, development of the theory of symmetry, namely the Group Theory had made it a useful tool for all branches of science, particularly for chemistry. Regular convex polyhedra, better known as Platonic Solids, which are only five in number, belong to the cubicfamily of Groups (Td, Oh and Ih). It serves as a mine of chemical information and produces the Gems of chemistry as/when extracted by the tool of Group Theory. The tentative point wise lay out of the talk is as follows:

1. Symmetry, Asymmetry and Anti symmetry

1.1 Symmetry as perceived since ancient times, 1.2. Asymmetry: The other side of the coin, 1.3. Had there been NO a) Symmetry, b) Asymmetry, 1.4. Ascent and Descent in molecular symmetry, 1.5. Asymmetry is not contradictory but complementary to Symmetry: Examples from Art and Music: Madonna Litta and Ek Taal, 1.6. Antisymmetry: Symmetry under exchange: The essence of JADIDANG HRIDAYANG TABA, TADIDANG HRIADAYANG MAMA

2. The Theory of Symmetry: Group Theory

2.1. Elements, Group Rules, Group, GMT, symmetry group, point group, 2.1. GMT: with example of a geometric figure and a simple molecule, 2.3. Mapping of a Group (G): Representatios, interrelation of, 2.4. Understanding reducible and irreducible (IR) representations, 2.5. Character Table: A demonstration using C3V and Td point groups.3. Platonic Solids: Its symmetry and chemistry

3.1. Definition and examples, 3.2. Playing Dice with an intelligent ET: How to win surely, 3.3. They are only five in number: Fortunate or unfortunate?, 3.4. Platonic solids and Sphere: Explaining the solar system, 3.5. Symmetry of Platonic solids4. Chemistry and Platonic Solids: A review with Td symmetry

4.1. The strategic lay out for using symmetry principles, in general, 4.2. A long list of use

of symmetry principles in chemistry, 4.3. A Case study with Td point group: a) Hybrid

Orbitals, b) Spectroscopic selection rule, 5. References

1.1 Symmetry as perceived since ancient times

Why? ▬► Phenomena «Law of Nature « Symmetry Principle ≈ GOD (!)

What (and Where)? ▬► Greek of “Proportionality, similarity in

arrangement of parts”

By an Ass: Equity, Balance (It starved to death!)*

By Man: Regular structure: Man made objects, for what? For Purpose ?*

Hermann Weyl: “… through symmetry man always tried to perceive

and create order, beauty and perfection”.

In Conjecture: Atom is a solar system in microscopic world (similarity)

From Nature: The Creator (Bramha, Prajapati)* and its creations …

Regularity in Day-Night cycle, cyclic seasons, Natural Laws,

Chhanda in poetry, Rhythm or harmony in music

Semi finally: Rabindranrth* [A song: Nrtyer taale taale, He Nataraj …]

Finally, Mathematical:

“… if an object(!) can be changed somehow to obtain the same object”

=> Unchangeability or invariance or conservation of some property (object or

phenomena) subject to certain transformation.

Few Natural Laws and Symmetry: Invariance of Physical Laws

Conservation of Invariance under Consequence of Example

Momentum Translations Homogeneity Newton’s

In space of space Laws

Angular Rotations in Isotropy of Kepler’s

Momentum space space Law of areas

Energy Translations Homogeneity 1st Law of

In time of time Thermodynamics

Mom.+Energy space & time Compton effect: Photon realized exptly

Additional Remarks:

Dimensionality in (space & time) and (momentum and energy).

Laws of Permission or Prohibition? 1st & 2nd tuned, Heisenberg, etc…

On One side of the coin: Symmetry

proportionality, similarity, equity, balance, regular, order, beauty and perfection,

chhanda or rhythm, harmony, unchangeability, invariance, conservation. . .Is there the

other side!

Nature is NOT Biased: CO2 + H2O Starch/sugars: asymmetry from symmetry

1.2. Asymmetry: The other side of the coin: ugly, disorder, variation?

Symmetry + Asymmetry

1. General Specific [Love at first sight]

2. Invariance Variation [1st 2nd Laws, column/row]

3. Repetition (performer) contrasted or Varied repetition [Ek Taal]*

4. Beauty Even more than that [Madonna ]*

5. ΔSsys < 0 (order) = Life ΔStotal ≥ 0 (disorder) = Life and death

*Beauty is a product of Marriage between symmetry and asymmetry

*As (life & death), so also (symmetry & asymmetry) are inseparable

* Thermodynamics gets completion

* Periodic table and hence the whole chemistry *A great Dialectic: Law of unity and

conflict of opposites

* Not contradictory but complementary:

# lessons (Buddhist)

Poison can be made into Medicine [motivation for chemists]

Make contradictory to complementary [Art of Living]

1.3. Had there been

NO Symmetry

1. Be satisfied with graphite, a far cry from the gem of gems: Diamond

2. Missing: 1st law of thermodynamics, periodic table, lots of chemistry, 3. Life: No more

worth living

NO Asymmetry

1. No “Through the Looking Glass” from Lewis Carroll,

2. Missing: 2nd law, periodic table, proteins & Nucleic acid, a few chapters

(spectroscopy, stereochemistry) form text (!), 3. Life: No more possible to live at.

1.4 & 1.5. Ascent and Descent in Molecular Symmetry*

Change of molecular properties as structure changes: molecular deformation or atomic

substitution.

Group theory: Group sub group relation can be used if basic geometry remains the same.

Note: Possible for h = 4, 8, 16, 48 => integer multiple

Not if h=8, 12 Not multiple.

A Musical Analogy: [Excuse Me !!] : Ek Taal* : An audible symmetry: Symmetry in time

Note: How Asymmetry blended with Symmetry Make a performer into an Artist.

1.6. Antisymmetry: Symmetry under exchange: The essence of

JADIDANG HRIDAYANG TABA, TADIDANG HRIADAYANG MAMA

Let a Heart be a beautiful, symmetric flower : ♥ or ψ

Two persons (I & II) with two Hearts (1 & 2): ψ1 & ψ2

ψ1(I) and ψ2(II) → Marriage → ψ1(I) ψ2(II)

Love = Sharing Hearts: So also to consider: ψ1(II) ψ2(I)

To combine the hearts and the shared hearts. But How? By (+) or (─)?

Ψ+ (I,II) = ψ1(I) ψ2(II) + ψ1(II) ψ2(I) Note: Ψ+ (I,II) = Ψ+(II,I)

Ψ− (I,II) = ψ1(I) ψ2(II) ─ ψ1(II) ψ2(I) Ψ− (I,II) = −Ψ−(II, I)

Which one is permitted? Or Both?

Max Born: A PURUT, knows commutative symmetry and a bit of quantum mechanics.

| Ψ (I,II)|2 = | Ψ (II,I)|2 [since (─)2 = (+), both Ψ+ and Ψ− are allowed]

Any difference between Ψ+ and Ψ− , any way?

Let them (I & II) be so close that both share the same heart, say, ψ1

Ψ+ (I,II) = ψ1(I) ψ1(II) + ψ1(II) ψ1(I) ≠ 0

Ψ− = ψ1(I) ψ1(II) ─ ψ1(II) ψ1(I) = Vanishes!!

Conclusion?

. Ψ + . . Ψ − .

1. symmetric 1. antisymmetric

2. 10 × 10, 10 2. likes to stay alone (Divorce!!)

3. collectivism 3. individualism

4. Bosons 4. Fermions

5. photons 5. electrons

6. otherwise: 6. otherwise:

no quantum mechanics no Pauli exclusion/Hunds rule

loss of versatility of chemistry

7. Gift: LASER etc. 7. versatility in elements

8. Molecular level: ortho 8. Molecular level: para

. .

*Caution!! Not to be strictly guided by the analogy in your conjugal life.

2.1. Elements, Group Rules, Group, GMT, symmetry group, point group

Group (G): Collection of members or elements that obey a set of Rules subject to a Law

of Combination (LoC) →(E, A, B, C, …..); order = h,

Elements: Numbers, figures, concepts, operations, etc.

LoC: AB → A+B or A×B or B followed by A [commutation, convention], etc.

Group Rules: G = (E, A, B, C, …, X, …)

R_1: Closure: AB = X infinity is not friendly

R_2: Associativity: ABC = A (BC) = (AC) B a perfect cooperative

R_3: Identity (E): AE = EA = A Apolitical or do nothing

R_4: Inverse: X X-1 = X-1X = E always anti

. 1, 2, 3, 4, … .

LoC = addition LoC = multiplication

R_1 & R_2 satisfied R_1,2,3 satisfied with E=1

R_3: X+E = E+X = X, E = 0 R_4: XX-1=E=1

R_4: XX-1 = X+X-1 = E = 0 X-1 = 1/X

X-1 = - X G = (+ve rational numbers, excl. 0)

G = (… -4 -3 -2 -1 0 1 2 3 4 …)

G0 ={1}, G1 = {1, -1}, G2 = {1, -1, i, -i } →LoC = multiplication, E=1

G, G1, G2 : Are they groups? Do they satisfy the 4 rules? To check?

To see all possible combinations of them, better through a TABLE (GMT)

Specific group Abstract group

1 -1 1 -1 i -i A B C .

1 | 1 -1 1 |1 -1 i -I A | AA AB AC .

-1 |-1 1 -1 |-1 1 -i I B | BA BB BC .

i |i -i, -1 1 C | CA CB CC . -i |-i i 1 -1 . | . . . .

*Convention: AB → Row (A) Column (B)

GMT defines G irrespective of the symbols used for gr. Elements

Many different sets of quantities (A B C ..) may have the same GMT

G can be represented by many of what we call REPRESENTATIONS (of G)

Symmetry Groups: Point Group (A Passage to Molecular world)

(A B C ..) = set of symmetry operations that follows the GMT

LoC → consecutive operation: [convention: AB: B followed by A]

[rotation, improper rotation, reflection, inversion]

2.2. GMT: with example of a geometric figure and a simple molecule

An equilateral Triangle : G = (E C3 C32 σ1 σ2 σ3 ) [h = 6]

GMT of a certain Abstract group satisfied by G, a specific group.

E itself a group (say G1) of h1 = 1 [like G0 = {1}]

(E σ1), (E σ2) and (E σ3); three smaller groups (G2) of h2 = 2 each

(E C3 C32 ); also a group (G3) of h3 = 3

G1, G2 and G3 are subgroups of group G

h = 6 is integer multiple of h1 (=1), h2 (=2) and h3 (=3)

(C3 C32 ) & (σ1 σ2 σ3 ): Two classes [common sense & S.T.]

Point Group:

Landed in Molecular world: From geometric figure to Molecular structure

A planar molecule with a structure of geometry of eq. Triangle

has the same set of symmetry operations that follows the same abstract GMT to

form the same G.

Naam? Not Hizibizibiz: Mulliken Nomenclature [here G = C3V]

Why called Point Group?

2.3. & 2.4. Mapping of G: Representatios, [reducible & irreducible(IR)]

Numbers are not only beautiful but also worth using than operations

collection of {1} satisfies any GMT, so (1 1 1 …) is an obvious representation of any G

G = (E C3 C32 σ1 σ2 σ3 ) [h = 6] →(imaging to) → G1=(1): 6:1 mapping

G = (E C3 C32 → 1 σ1 σ2 σ3 → -1) →→ G1 (1 -1) 3:1 mapping

Let each R of G be replaced by different matrix [M] (say, 2×2) and

the collection of {M} satisfies the GMT of group G.

G = (E C3 C32 σ1 σ2 σ3 ) →→ G3 (Mk )k=1,6 1:1 mapping

G | E C3 C32 σ 1 σ 2 σ 3

G1| 1 1 1 1 1 1

G2| 1 1 1 -1 -1 -1 *G4| χ1 χ2 χ3 χ4 χ5 χ6

G3| M1 M2 M3 M4 M5 M6 χM = ∑ M(i,i)

How many such Gis are possible? → infinite (what the hell!)

Are all meaningfully different? → Fortunately NOT

Most of them (reducible) are sum of other fundamental, simplest representation

(irreducible) of lower dimension

If not, how are they related? → similarity transformation

Reducible ones are but not the IRs.

e′ = e T, R′ = T-1RT, S′ = T-1ST, Q′ = T-1QT, RS = Q, R′S′ = Q′

e′ → {ei′} and e → {ei }; e and e′ are basis, here unit vectors

Matrix Representation ({M}): What really the matrices do? How are they constructed?

R ● → equivalent ●; R ↑ → eq. ↑; (symbol) ↑ → eq. ↑ = [x y z ]

M (A suitable matrix) [x y z] → equivalent [x y z]

Let R = c2z R [x y z] = c2

z [x y z] = [-x –y z] M [x y z] = [-x –y z]

|-1 0 0| | x | |-x | |1 0 0| | x | |x |M(c2

z) = |0 -1 0| | y | =|-y | M(E) = |0 1 0| | y | = |y|χ (c2

z) = -1 |0 0 1| | z | | z | χ (E)= 3 |0 0 1| | z | |z |

We get the reducible representations {M(R)} and χ (R)

2.5. Understanding Character Table: using C3V and Td point group (G)

C3V E 2C3 3σv

A1 1 1 1 z x2 + y2, z2

A2 1 1 -1 Rz

E 2 -1 0 (x, y), (Rx, Ry) (x2 - y2, xy), (xz, yz)

. .

Td E 8C3 3C2 6S4 6σd

A1 1 1 1 1 1 x2 + y2 + z2

A2 1 1 1 -1 -1

E 2 -1 2 0 0 (2z2 - x2 - y2, x2 - y2)

T1 3 0 -1 1 -1 (Rx, Ry, Rz)

T2 3 0 -1 -1 1 (x, y, z) (xy, xz, yz). .

Gives the complete list of the IRs , their symmetry identity (1st col)

Each row is a group, satisfies the GMT of the concerned G

How many such IRs? = number of classes (nc); =3 (C3V), =5 (Td)

IRs are nc-dimensional vectors [GOT], nc independent vectors

χ IR (R) : entries in the row: character of IR under R (2nd col)

Transformation of linear (x, y, z) and rotational (Rx,Ry, Rz) vectors

Direct products of linear vectors [z2, x2 – y2, xy, yz, xz] → d-orbitals

Two or more vectors may transform as the same symmetry species but may not be

degenerate. Parentheses are missing x, y

A Few more points for a Chemist

See 3rd col for (x, y) in the E-row (IR) in C3V [(x, y, z) in T2 in Td]

X and Y transform as a degenerate pair. No difference between X and Y directions.

If a property along X-direction, there will be an equivalent and indistinguishable property

along Y. Px and py orbitals on a central atom have same energy, indistinguishable.

See 3rd column for z in the A1-row in C3V

Unit vector Z transforms as A1. Transformation of or (property along) Z direction is

totally symmetric and is different from those of E symmetry [(x, y)]. Pz orbital is of

different energy and be distinguishable from the degenerate pair of px and py orbitals.

Placing an atom in a C3V environment lifts the threefold degeneracy of the p orbitals

unlike differently from that in Td environment.

3.Platonic Solids: Its symmetry and chemistry

3.1. Definition and examples*: 3.5. Its symmetry

Regular Polyhedron

Faces are all equivalent and are all regular polygon [say, n-gone, n ≥ 3,

equilateral triangle (n=3), square (n=4), regular pentagon (n=6), etc.]

Vertices are all equivalent; Edges are all equivalent

Examples of regular polyhedra:

Polyhedron Polygon face Vertices Edges h

Tetrahedron Triangle 4 4 6 24

Cube square 6 8 12 48

Octahedron Triangle 8 6 12 48

Dodecahedron Pentagon 12 20 30 120

Icosahedron Triangle 20 12 30 120

(2 & 3) are mutual and (4 & 5) are mutual [with same h]

Note: face and vertices are also mutual.

24, 48, 120 , multiples of 12: inspiration for Satyajit Ray?

3.2. Playing Dice with an intelligent ET: How to win surely

“Inhabitant of even the most distant galaxy can not play dice having a shape of a regular

convex polyhedron unknown to us.” → Martin Gardner

The n-gons or faces meet at a point (vertex) to form a closed, non-planar arrangement.

How many (say, m, m ≥ 3) polyhedra are possible with n-gon face?

If θ = each angle of a n-gon, θ = (2 n - 4) π/(2n)

The sum of the angles (m θ) around the vertex < 3600, otherwise planar

n face θ m for (m θ) < 3600 Polyhedron

3 triangle 600 m=3, (m θ) = 1800 Tetrahedron (Td)

m=4, (m θ) = 2400 Octahedron (Oh)

m=5, (m θ) = 3000 Icosahedron (Ih)

m=6, (m θ) = 3600 Not Possible

4 square 900 m=3, (m θ) = 2700 Cubic

m=4, (m θ) = 3600 Not Possible

5 pentagon 1080 m=3, (m θ) = 3240 Dodecahedron

m=4, (m θ) = 4320 Not Possible

≥ 6 ≥ hexagon 1200 m ≥ 3, (m θ) ≥ 3600 Not Possible

3.3. They are only five in number: Fortunate or unfortunate?

Symmetry restriction on variety of structures: only 14 Bravais Lattice

3.4. Platonic solids and Sphere: Explaining the solar system

Harmony of the creation of the World: sign of divine will ancient scholars

As Platonic solids are at the heart of structure of matter and the Universe

(Not so) ancient Belief: 4 principal elements of matter have the shape of Platonic solids:

fire→tetrahedra, earth → cubic air → octahedra water→icosahedra

Kepler (1571-1630): six planets (O1-6, sphere) in the solar system

Six ‘Dots’ on a chain with 5 gaps or links which are those 5 Platonic solids,

one inscribed into other.

O1→Cube→O2→tetrahedron→O3 → dodecahedron → O4 → icosahedron→ O5 →

octahedron → O6

Ratio of radii of the spheres: 8:15:20:30:115:195 : Excellent agreement

Kepler: “… my hypothesis was in agreement with Copernicus’s theory and my JOY did

not vanish into smoke”

4.1. The strategic lay out for using symmetry principles: A six step package

Step I: Find the symmetry elements, symmetry operations i.e. the group elements, classes

of them and hence the point group (G) of the molecule with known/proposed structure.

Step II: Choose a suitable basis, usually unit vectors at all the atoms or the orbitals or

bonds or normal modes etc.

Step III: Find the reducible representation in terms of characters (of marices)

Step IV: Decompose the red. Representations into its IRs, find their numbers

Step V: Consult the Character Table (of G) for 3rd and 4th column for finding the

symmetry of the IRs and hence important chemical information.

Step VI: Compare the results with the experimental ones

4.2. A long list of use of symmetry principles in chemistry

1. Selection rules in spectroscopy, mutual exclusion principle

2. symmetry of normal modes of vibration: IR spectroscopy

3. Hybridization of atomic orbitals: A gateway to modern chemistry

4. Molecular orbital theories: Chemical bonding

5. Symmetry Adapted Linear Combinations (SALC) of MOs

6. Crystal Field Theory (CFT or LFT)

7. Term splitting, 8. …, 9. ……

4.3. A Case study:Td point group: a) Hybrid Orbitals*, b) Spectros. selection rule*

Hybrid orbitals: Pauling

“I would prefer to be remembered as the man of hybridization rather than the man of

Vitamin C” [he was often called as the man of vitamin C]

Basis: 4 equivalent hybrid orbitals as vectors (A, B, C, D), Classes: (E C3 C2 S4 σd )

The matrices M(R):

|1 0 0 0| | A | |A | |1 0 0 0| | A | |A |

M(E) = |0 1 0 0| | B | = |B | M(C3 ) = |0 0 0 1| | B | = |D |

χ (E) = 4 |0 0 1 0| | C | |C | χ (E) = 1 |0 1 0 0| | C | |B |

|0 0 0 1| | D | |D | |0 0 1 0| | D | |C |

|0 1 0 0| | A | |B | |0 0 1 0| | A | |C |

M(C2) = |1 0 0 0| | B | = |A | M(S4 ) = |0 0 0 1| | B | = |D |

χ (E) = 0 |0 0 0 1| | C | |D | χ (E) = 0 |0 1 0 0| | C | |B |

|0 0 1 0| | D | |C | |1 0 0 0| | D | |A |

Td E 8C3 3C2 6S4 6σd (h=24)

A1 1 1 1 1 1 x2 + y2 + z2

A2 1 1 1 -1 -1

E 2 -1 2 0 0 (2z2 - x2 - y2, x2 - y2)

T1 3 0 -1 1 -1 (Rx, Ry, Rz)

T2 3 0 -1 -1 1 (x, y, z) (xy, xz, yz)

χred 4 1 0 0 2

nIR = (1/h) ∑ gc χIR χred; nIR = number of occurrence of the IR in Red. Reprsn.

Td E 8C3 3C2 6S4 6σd (h=24)

A1 1 1 1 1 1 x2 + y2 + z2

T2 3 0 -1 -1 1 (x, y, z) (xy, xz, yz)

χred 4 1 0 0 2

Гred = A1 + T2, A1 → totally symmetric → s orbital, T2 → (xy, xz, yz) → 3 degenerate d-

orbitals, so it involves one s and 3 d-orbital, a sd3 hybridization (surprised?).

Also T2 → (x, y, z) → 3 p-orbitals, so involves one s and 3 p-orbital, a sp3 hybridization.

Though surprising, symmetrically both sp3 and sd3 hybridization are allowed.

However, due to energy gap factor, sp3 is the dominating one. But sd3 is not so rare. It is

sp3 that explains CH4 or CCl4 molecules but if the central metal atom is a transition metal

(CrO42-), sd3 is significant.

For a linear (D∞h) molecule, both sp and dp hybridization are allowed symmetrically.

4.4. A Case study with Td point group: b) Spectroscopic selection rule*

Quantum mechanics:

vibrational transition in the Infrared or Raman spectrum: Intensity is nonzero if the

corresponding transition moment (M) is nonzero. M(0,1) = ∫ψ0 μ ψ1 dτ ψ0 and ψ1 are

the wave functions

μ→ oscillating electic dipole moment vector having components μx, μy , μz

Mx = ∫ψ0 μx ψ1 dτ My = ∫ψ0 μy ψ1 dτ Mz = ∫ψ0 μz ψ1 dτ

M(0,1) is nonzero if any one of the above components is nonzero.

Symmetry :

∫fA fB dτ is nonzero if it is invariant under all R of G. Possible only if (fA fB) is or

contains totally symmetric representation of G.

ψ0 [ground state] of all molecule is totally symmetric

ψ1 has the symmetry of the normal mode.

A1 × XIR = XIR → (ψ0 ψ1) = ψ1

ГIR × ГIR (self product) = is or contains totally symmetric represtn.

Thus, if any component of μ has the same symmetry as ψ1, the product (μ ψ1) will

be totally symmetric and the integral is non vanishing

Vector components μx, μy, μz transform as unit vector x, y, z respectively.

A normal mode with the symmetry of any of the unit vector transformation x, y or z

will be active in the infrared spectrum.

Task to find infrared active modes, to find the symmetry of the normal modes, to

look at the unit vector symmetries in the character table

A1 E 2T 2 For: Td XY4 molecules

ν1 ν 2 ν 3 ν4

Raman Raman IR and Raman

superscript* indicates the corresponding figures or diagrams which are not included here.

5. References

1. Symmetry: Weyl Hermann, Princeton University Press (1952)

2. The Equation that could not be solved: How mathematical Genius Discovered

the Language of Symmetry: Livio Mario (2005), N. Y.

3. This Amazingly Symmetrical World: L. Tarasov, MIR, Moscow

4. Symmetry- An Introduction to Group Theory and Its Applications: Roy

McWeeny, Dover Pub., New York

5. Chemical Applications of Group Theory: F.A. Cotton

6. Group Theory and Quantum Mechanics: Tinkham, McGraw-Hill

7. Group Theory and Chemistry: D. M. Bishop, Dover Pub., N.Y.

8. Molecular Symmetry and Group Theory: R. L. Carter, John Wiley & Sons

9. A Lifelong Quest For Peace* : A Dialogue: Linus Pauling and Daisaku Ikeda,

I.B. Tauris, London, N.Y.

10. http://www.adrianbruce.com/symmetry

11. http://gwydir.demon.co.uk (for on-line playing with symmetry pattern).

A couple of symmetrical words blended with a bit of asymmetry

T

H

A

N

K

Y

O

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