# Symmetries of the alphabet.

The letters of the alphabet exhibit five varieties of symmetry (or
non-symmetry):

A M T U V W Y have left-right (vertical) symmetry (.272) v B C D E K have up-down (horizontal) symmetry (.218) h N S Z have 180[degrees] (rotational) symmetry (.137) r H I O X have all three symmetries (.205) a F G J L P Q R have no symmetry (,168) n

In running text, thee various symmetries appear with the indicated probabilities; for example, left-right symmetry occurs twice as often as rotational symmetry,

These appear to be all the possible theoretical symmetries. In particular, it is impossible to create a "letter" with two but not three symmetries, i.e., with left-right and rotational (but not up-down), with up-down and rotational (but not left-right), or with left-right and up-down (but not rotational) symmetry.

Can one locate a set of 120 five-letter heterograms which contain one letter with each kind of symmetry, arranged in all possible symmetry orders? Using only words in boldface from Webster's Second or Third (plus inferred forms such as past tenses and plurals), it appears to be impossible. One difficulty appears to be that vowels are contained in only 3 of the 5 symmetries. If one allows OED words or placenames, however, then a full collection is likely possible.

The collection below is restricted to Websterian words. Each possible ordering is identified by lower-case letters vhran. Although only a couple of examples are typically shown, a few orders are especially fecund:

navhr GIVEN LIVEN RIVEN GIVES LIVES FIVES RIVES LIMES RIMES RITES LOVES LOADS ROADS GOALS hnavr CLOYS CLOTS BLOTS BLOWS BROWS CROWS BLOWN BROWN CROWN CLOWN BRIMS DROWN rvanh SMILE SMIRK SHORE SWORD STILE SWIPE STOLE STOPE STORE STORK

ahnrv

ahnvr HEFTS

ahrnv HENRY

ahrvn

ahvnr HEAPS HEALS

ahvrn

anhrv OLENT

anhvr OREAS

anrhv

anrvh

anvhr OFTEN

anvrh

arhnv INEPT INERT

arhvn INCUR

arnhv INLET

arnvh INRUB

arvhn INTER

arvnh INURE

avhnr OWERS

avhrn

avnhr HARES

avnrh HALSE

avrhn OWNER HAZEL

avrnh HAZLE

hanrv CORNY

hanvr KORAN BOGUS

harnv CONGA DINGY

harvn DINAR

havnr BOARS CHAPS

havrn BHANG

hnarv BRINY KRONA

hnavr BROWN CLOTS

hnrav EPSOM

hnrva

hnvar DRAIN BLITZ

hnvra BRUSH CLASH

hranv

hravn

hrnav ENJOY

hrnva

hrvan ESTOP

hrvna

hvanr BAIRN CAIRN

hvarn DYING

hvnar BARON CAPON

hvnra DARSO

hvran BASIL

hvrna BNJO

nahrv POESY

nahvr FOCUS LOCUS

narhv FONDU RISKY

narvh GONAD

navhr GIVEN LOVES

navrh FOUND PHASE

nharv FEINT PEONY

nhavr REOWN

nhrav RESOW

nhrva PESTO LENTO

nhvar LEMON

nhvra LEASH

nrahv

nravh GNOME

nrhav

nrhva

nrvah

nrvha PSYCH

nvahr RAIDS

nvarh GUISE

nvhar RADON

nvhra

nvrah PANIC RUNIC

nvrha LYNCH JUNCO

rahnv NOBLY

rahvn SHEAF SIBYL

ranhv SHREW SILKY

ranvh SHRUB SOLVE

ravhn NITER SOWER

ravnh SHAPE SHARK

rhanv SKIRT

rhavn SCOWL SKIMP

rhnav SCRIM SEPIA

rhnva SERMO SEPTI

rhvan NETOP

rhvna SCAPI

rnahv SPICY SPIKY

rnavh SLIME SPITE

rnhav

rnhva

rnvah SQUIB SQUID

rnvha

rvahn STOEP

rvanh SMILE SMIRK

rvhan NADIR

rvhna SACRO

rvnah SALIC

rvnha

vahnr TIERS

vahrn

vanhr WORDS MILKS

vanrh WORSE

varhn WISER

varnh AISLE TINGE

vhanr ACORN ADORN

vharn

vhnar AEGIS MELON

vhnra VERSO WELSH

vhran TENOR

vhrna

vnahr ALIEN TRIES

vnarh ALONE ARISE

vnhar VLEIS

vnhra

vnrah

vnrha

vrahn USHER

vranh ANILE

vrhan UNDOG

vrhna

vmah ASPIC

vrnha

Finally, a challenge to computer programmers: can the alphabet be partitioned into five groups so that all 120 permutations can in fact be found among Websterian heteronyms?

A. ROSS ECKLER

Morristown, New Jersey

A M T U V W Y have left-right (vertical) symmetry (.272) v B C D E K have up-down (horizontal) symmetry (.218) h N S Z have 180[degrees] (rotational) symmetry (.137) r H I O X have all three symmetries (.205) a F G J L P Q R have no symmetry (,168) n

In running text, thee various symmetries appear with the indicated probabilities; for example, left-right symmetry occurs twice as often as rotational symmetry,

These appear to be all the possible theoretical symmetries. In particular, it is impossible to create a "letter" with two but not three symmetries, i.e., with left-right and rotational (but not up-down), with up-down and rotational (but not left-right), or with left-right and up-down (but not rotational) symmetry.

Can one locate a set of 120 five-letter heterograms which contain one letter with each kind of symmetry, arranged in all possible symmetry orders? Using only words in boldface from Webster's Second or Third (plus inferred forms such as past tenses and plurals), it appears to be impossible. One difficulty appears to be that vowels are contained in only 3 of the 5 symmetries. If one allows OED words or placenames, however, then a full collection is likely possible.

The collection below is restricted to Websterian words. Each possible ordering is identified by lower-case letters vhran. Although only a couple of examples are typically shown, a few orders are especially fecund:

navhr GIVEN LIVEN RIVEN GIVES LIVES FIVES RIVES LIMES RIMES RITES LOVES LOADS ROADS GOALS hnavr CLOYS CLOTS BLOTS BLOWS BROWS CROWS BLOWN BROWN CROWN CLOWN BRIMS DROWN rvanh SMILE SMIRK SHORE SWORD STILE SWIPE STOLE STOPE STORE STORK

ahnrv

ahnvr HEFTS

ahrnv HENRY

ahrvn

ahvnr HEAPS HEALS

ahvrn

anhrv OLENT

anhvr OREAS

anrhv

anrvh

anvhr OFTEN

anvrh

arhnv INEPT INERT

arhvn INCUR

arnhv INLET

arnvh INRUB

arvhn INTER

arvnh INURE

avhnr OWERS

avhrn

avnhr HARES

avnrh HALSE

avrhn OWNER HAZEL

avrnh HAZLE

hanrv CORNY

hanvr KORAN BOGUS

harnv CONGA DINGY

harvn DINAR

havnr BOARS CHAPS

havrn BHANG

hnarv BRINY KRONA

hnavr BROWN CLOTS

hnrav EPSOM

hnrva

hnvar DRAIN BLITZ

hnvra BRUSH CLASH

hranv

hravn

hrnav ENJOY

hrnva

hrvan ESTOP

hrvna

hvanr BAIRN CAIRN

hvarn DYING

hvnar BARON CAPON

hvnra DARSO

hvran BASIL

hvrna BNJO

nahrv POESY

nahvr FOCUS LOCUS

narhv FONDU RISKY

narvh GONAD

navhr GIVEN LOVES

navrh FOUND PHASE

nharv FEINT PEONY

nhavr REOWN

nhrav RESOW

nhrva PESTO LENTO

nhvar LEMON

nhvra LEASH

nrahv

nravh GNOME

nrhav

nrhva

nrvah

nrvha PSYCH

nvahr RAIDS

nvarh GUISE

nvhar RADON

nvhra

nvrah PANIC RUNIC

nvrha LYNCH JUNCO

rahnv NOBLY

rahvn SHEAF SIBYL

ranhv SHREW SILKY

ranvh SHRUB SOLVE

ravhn NITER SOWER

ravnh SHAPE SHARK

rhanv SKIRT

rhavn SCOWL SKIMP

rhnav SCRIM SEPIA

rhnva SERMO SEPTI

rhvan NETOP

rhvna SCAPI

rnahv SPICY SPIKY

rnavh SLIME SPITE

rnhav

rnhva

rnvah SQUIB SQUID

rnvha

rvahn STOEP

rvanh SMILE SMIRK

rvhan NADIR

rvhna SACRO

rvnah SALIC

rvnha

vahnr TIERS

vahrn

vanhr WORDS MILKS

vanrh WORSE

varhn WISER

varnh AISLE TINGE

vhanr ACORN ADORN

vharn

vhnar AEGIS MELON

vhnra VERSO WELSH

vhran TENOR

vhrna

vnahr ALIEN TRIES

vnarh ALONE ARISE

vnhar VLEIS

vnhra

vnrah

vnrha

vrahn USHER

vranh ANILE

vrhan UNDOG

vrhna

vmah ASPIC

vrnha

Finally, a challenge to computer programmers: can the alphabet be partitioned into five groups so that all 120 permutations can in fact be found among Websterian heteronyms?

A. ROSS ECKLER

Morristown, New Jersey

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Author: | Eckler, A. Ross |
---|---|

Publication: | Word Ways |

Geographic Code: | 1USA |

Date: | Aug 1, 2007 |

Words: | 572 |

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