Clark Kimberling

Professor of Mathematics, University of Evansville, 1800 Lincoln Avenue, Evansville, IN 47722
Welcome to my home page. Here you'll find various mathematical, historical, and musical topics, in many cases matched with biographical sketches. The mathematical topics include triangle geometry and number theory. The section on triangle geometry contains the world's largest internet collection of triangle centers. The number-theoretic topics deal mostly with integer sequences and arrays.

The historical collection started out with 19th century scientists associated with the famous town, New Harmony, Indiana, located about 30 miles from Evansville. The collection has grown to include educators, writers, and artists.

The music section consists of projects, including links to free downloadable scores and recordings. Ongoing projects include arranging and composing for groups of recorders. Below, you can click and listen to 75 duets and 117 trios performed by Japanese recorder artist Papalin.

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ENCYCLOPEDIA OF TRIANGLE CENTERS - ETC
Triangle Centers: Classical and Recent
Euler Line, including 102 triangle centers
Algebraic Highways in Triangle Geometry
Triangle Centers and Central Triangles (book)


INTEGER SEQUENCES AND ARRAYS
Unsolved Problems and Rewards
Interspersions and Dispersions
Continued Fractions
Fractal Sequences


BIOGRAPHICAL STUDIES
New Harmony Scientists, Educators, Writers & Artists
Emmy Noether, Her Mentors & Colleagues
Fibonacci Number-Theorists
Triangle Geometers


MUSIC PROJECTS

Project 1. Solos for Treble Instrument, Especially Soprano Recorder
This completed project consists of 12 collections, each in two parts: Historical Notes and Solos. You can click below to access the Historical Notes, and from there, you can download most of the Solos. The others are available from Mel Bay Publications.

Collection 1: African-American and Jamaican Melodies
Collection 2: Christmas Carols (available from Mel Bay Publications.)
Collection 3: Irish Melodies
Collection 4: Americana to 1865
Collection 5: Americana after 1865
Collection 6: British Melodies
Collection 7: Melodies by Women Composers
Collection 8: Eastern European and Jewish Melodies
Collection 9: American Indian Melodies
Collection 10: Latin American Melodies
Collection 11: African Melodies
Collection 12: Western European Melodies

Project 2. Compositions for church choirs, available from G.I.A. Publications:

Four Anthems for Mixed Voices and Handbells
Scripture and Season in Song
The Hills Are Hushed This Night of Nights
O God, Beneath Your Hand
O God, Who at the Dawn of Time
Ring Out the Glad Tidings
The King of Love My Shepherd Is
This Easter Morn
Awake with Timbrel and with Dance
Introits and Anthems for Voices and Bells, Volume 1
Introits and Anthems for Voices and Bells, Volume 2
Introits and Anthems for Voices and Bells, Volume 3

Project 3. New hymn tunes, with texts by James Hart Brumm, John Core, Ralph Freeman, Daniel P. Merrick, and John Thornburg. The book is available from Wayne Leupold Editions; click Information.

Project 4. Henry Hart, a composer who lived in Evansville, Indiana (where I live) in the 1870's. Hart published minstrel tunes and music for social dance. The research led to the first of my biographical articles for Wikipedia: Henry Hart. The External Links section at the end will take you to three supplements:

Supplement I. Henry Hart and His Family Orchestra (Including Eli Lilly's Recollections)
Supplement II. Hazel Hart Hendricks and Her School
Supplement III. Myrtle Hart, Harpist

Project 5. Articles in The Hymn, the journal of The Hymn Society in the United States and Canada:

Tunes Lost and Found: BIRSTAL, FOUNDRY, HOTHAM, TRUE ELIJAH, The Hymn 52, no. 1 (Jan., 2001) 21-30.

Tunes Lost and Found: HELMSLEY, LEEDS, YORKSHIRE, WEDNESBURY, The Hymn 52, no. 2 (Apr., 2001) 25-36.

Tunes Lost and Found: TRUMPET, LOVE-FEAST, ATHLONE, DERBY, The Hymn, no. 2 (Apr., 2002) 7-18.

The Hymn Tunes of Richard Wayne Dirksen, The Hymn 53, no. 4 (Oct., 2002) 19-28.

Hymn Tune Descants, Part 1: 1915-1934, The Hymn 54, no. 3 (July, 2003) 20-27. Reprinted in Journal of the Ralph Vaughan Williams Society 29 (Feb., 2004) 17-20.

Hymn Tune Descants, Part 2: 1935-2001, The Hymn 55, no. 1 (Jan., 2004) 17-22.

Three Native American Hymns, The Hymn 56, no. 2 (Spring 2005) 18-29. Regarding these hymns, see the Wikipedia article Joseph Renville.

The Hymn 64, no. 3 (Summer 2013) 7-18.

Project 6. Christmas carols for flute with keyboard accompaniment: Tidings of Comfort and Joy.

Project 7. Duets for Flutes or Recorders for flutes, recorders (bass, tenor, alto, soprano, sopranino), or other instruments: DUETS - Publisher's description.

The duets are performed on various pairs of recorders, from contrabass up to sopranino, by Papalin:
Part 1: Duets 1-27
Part 2: Duets 28-50
Part 3: Duets 51-75

Project 8. Trios for Recorders or Flutes. For downloadable free scores, click Online Sheet Music: Recorder Trios.

The 117 trios in this collection are performed on various recorders, from contrabass up to sopranino, by Papalin:
Part 1: Trios 1-27
Part 2: Trios 28-58
Part 3: Trios 59-90
Part 4: Trios 91-117

For access to the vast collection of Papalin's recordings, IL DIVO "Papalin"

Project 9. Quartets for Recorders. Scored for various ensembles of sopranino, soprano, alto, tenor, and bass recorders. For downloadable free scores, click Online Sheet Music: Recorder Quartets.


ARTICLES IN THE CANTERBURY DICTIONARY OF HYMNOLOGY

The Canterbury Dictionary of Hymnology was launched from the University of Bristol in 2013. See Home Page and Editors' Introduction. My interest in hymn-tune descants, found in 1915 in England with not-yet-found French precedents, led to an article in the Journal of the Ralph Vaughan Williams Society, and from there to the Descant article in the CDH. Subsequently, I was invited to write about American hymn tune composers and early American hymn tune collections, such as William Walker's Southern Harmony. Following is a list of the resulting articles:

Lee Hastings Bristol, Jr., 1923-1979
Allen Carden, 1792-1859
Amzi Chapin, 1768-1835
Christian Harmony, 1867 -
The Colored Sacred Harp, 1934 -
Elkanah Kelsay Dare, 1782-1826
Ananias Davisson, 1780-1857
Descant, 1915 -
Richard Dirksen, 1921-2003
Stephen Foster, 1826-1864
Philip Frazier, 1892-1964
Louis Moreau Gottschalk, 1829,1869
Julia H. Johnston, 1849-1919
Kentucky Harmony and Supplement,, 1816 -
Elisha J. King, 1821?-1844
Missouri Harmony, 1820 -
John Jacob Niles, 1892-1980
Charles S. Nutter, 1842-1928
Amos Pislbury, 1772-1812
Shenandoah Harmony, 2012
George Frederick Root, 1812-1895
The Sacred Harp, 1844 -
Southern Harmony, 1835 -
Timothy Swan, 1758-1843
William Walker, 1809-1875
Benjamin Franklin White, 1800-1879
Frederick Jerome Work, 1878?-1942
John Wesley Work (I), 1848?-1923
John Wesley Work (II), 1872?-1923
John Wesley Work (III), 1901-1967
Wyeth's Repository of Sacred Music, 1813 -


ONLINE ENCYCLOPEDIA OF INTEGER SEQUENCES

Neil Sloane's Online Encyclopedia of Integer Sequences (OEIS) is a treasury of mathematical patterns, including references, links, and comments. Many contributors have developed prototypical sequences that introduce newly discovered relationships among numbers, some proved and others conjectural. Here are several:

Accumulation arrays and chains, weight arrays: A144112, A185906, A185778
Adjective-noun sequences and conjectures: A055168- A055185 and A055186- A055191
Adjusted joint rank sequences: A186219, A186221, A186350
Augmentation arrays: A193091
Averaging arrays: A197737
Beatty arrays: A181661, A181886, A183988
Beatty discrepancy, complementary equations: A136496, A138253, A138330
Change-of-base functions (geometric fractals): A214969
Constant associated with fractal change-of-base functions: A188593,
Constants associated with special shapes: A188593, A188618, A188635, A195284 (Philo distance, incenter) A195304 (Philo distance, centroid)
Constants associated with extreme values: A196500
Constants associated with convergent sums of reciprocals: A228040, A228044
Constants associated with tangencies: A196625, A197016
Constants that solve transcendental equations: A196396, A196401, A196407, A197737, A197739
Continued fractions, accelerated: A228667
Continued fractions, delayed: A228825
Continued fractions (including conjectures): A013659 (initialized continued fractions), A096654- A096658 (self-generating continued fractions), A143515, A143527, A229287 (primes), A229348 (Fibonacci-ratio terms), A229351 (fractional terms), A229981 (associated with divergent continued fraction), A229983 (associated with divergent continued fraction), A229985 (associated with divergent continued fraction), A229987 (associated with divergent continued fraction), A230000 (continued fraction [1,1/x,1/x^2,...])
Continued fraction transform: A229594
Convolution arrays: A213500
Convolutory inverses: A077607, A225127
Dispersions: A114537 (the primes), A191426
Divided difference sequences: A204892
Double interspersion: A161179
Egyptian fractions (denominates, convergent series): A226049
Euclidean algorithm, modified, accelerated (see also Continued fractions, accelerated, delayed): A228247, A228468 (definition of trace of (x,y)), A228486 (definition of trace of a number, in particular, pi), A228487, A228488
Iteration from natural numbers to lower Wythoff sequence: A181108
Fission arrays (of polynomial sequences, infinite matrices) A193842, A193844 (Pascal self-fission array) A202503 (Fibonacci self-fission array)
Floor-sum sequence: A182653
Fractal sequences: A003602, A194029, A194832
Fractalization: A194959
Fractional parts (counts involving inequalities, intervals): A194285 A194738, A194813
Fractional parts (difference between): A194368
Fusion arrays (of polynomial sequences, infinite matrices) A118800 (Pascal self-fusion array), A193722, A202453 (Fibonacci self-fusion array)
Golden triangle: A152149, A178988, A188635
Harmonic mean: A227012 (includes conjecture)
Harmonic numbers: A088907, A224820, A224868, A225918 (partial sums of harmonic series), A22a6894, A227581, A227631 (definition of least splitter, a conjecture)
Joint-rank arrays: A184912- A184927, A182801, A182869
Jointly generated arrays (by paired polynomial recurrences): A208510
Jointly ranked sequences (with a conjecture): A206911
Jump sequences (involving floor and ceiling, with conjectures): A219085
Kimberling expulsion array: A006852, A007963, A035486
Knight's moves on infinite chessboard: A183049
Kolakoski fan conjectures: A144586, A143589
Lagrange pairs: A194508
Lattice points in a region, counted: A227347
Limiting counting procedure (Theorems 1 and 2): A225660, A225669
Matrix counts: A194508
Morphism sequences: A189576, A189628
Mysterious B-sequence (array): A186158
Natural number array: A000027, A185787
Number of diophantine solutions (w,x,y,n), etc.: A211266, A211422, A211795, A212959
Order arrays: A163254, A163255
Ordering of rational numbers and other sets; see also "Trees generated by various sequences" and "Rules-generated sequences" : A226080 (Fibonacci - or rabbit - ordering, all positive integers), A226130 (all rationals), A226247 (all rationals), A232559 (all positive rationals), A232723 (all integers), A232895 (all positive integers; linear recurrence conjecture), A232890 (all rationals), A233694 (all Gaussian rationals)
Palindromic linear recurrences; array R(n,k) = n+[n/2]+...[n/k]: A211701
Partial complements, polygonal numbers: A086270, A086272
Partitions of the positive integers: A184812, A209082 (least power separator), A209122, A209354, A209423, A209489
Partitions of the set of positive integers (by joint rankings): A184928
Permutations of integers (conjectures): A131388- A131395
Polynomial reduction: A192175, A192232, A192744, A192872, A192904, A192951
Power ceiling functions: A214992 (several open questions)
Primes (conjectures): A174349, A184774, A185510, A219096
Prime divisor subtractions iterated until reaching 0: A233341
Prime-related arrays: A192175, A229608
Pythagorean fraction approximations: A195500
k-Pythagorean triples: A195770
Q-residues of polynomials: A193649
Rank transforms: A187224
Repetition-resistant sequence: A007061, A079101
Rules-generated sequences: A074049, A147991, A183544, A190803, A191106
Self-generating arrays: A203300, A203301
Shift-sums of zero-one sequences: A187950, A188014
Signature sequences: A007337, A084531
Symmetric matrices and interlacing roots of polynomials: A202605, A204016
Symmetric matrices and permanents: A204263, A204269, A204435, A204551
Trees generated by rules about edges: A228853
Trees generated by various sequences: A074049, A183079, A183169, A183231, A183232, A183542, A233135 (shortest (x+1,2x)-code of n), A232895
Unimodular 2x2 matrices: A210000
Vandermonde permanents: A203468
Wythoff array, W: A035513, A114579 (transposition sequence of W), A185778, A186007
Wythoff difference array: A080164
s-Wythoff sequences: A184117
Zeckendorf distance between m and n: A226207
Zero-one sequences (self-generating): A188967


GALLERY OF SPACE CURVES (Click!)

SECOND GALLERY OF SPACE CURVES (Click!)

A pdf of The Shape and History of The Ellipse in Washington, D. C. (Click!)