Uddiṣṭa: Another concise algorithm of Piṅgalācārya

India has been a cra­dle not only to refined civ­i­liza­tion­al best-prac­tices but also to mul­ti­tudi­nous sci­en­tif­ic devel­op­ments. Evi­dence and knowl­edge of volu­mi­nous lit­er­a­ture pro­duced in Indi­an sci­en­tif­ic pur­suits has been well estab­lished by seri­ous researchers. In con­trast, there is preva­lent igno­rance about facts and feats that we have inher­it­ed.

Through a series of short arti­cles, we have set forth to cov­er a few high­lights in the devel­op­ment of Jyotiṣa (Astron­o­my) and Gaṇitā (Math­e­mat­ics). The devel­op­ment of both these fields can broad­ly be put in three eras:

1. Vedic or Pre-Sid­dhan­tic Era,

2. Sid­dhan­tic or Clas­si­cal Era and

3. Post-Sid­dhan­tic or Medieval Era.

Hav­ing delved into the Vedic peri­od, we have moved ahead in the time­line and pre­sent­ed an intro­duc­to­ry glimpse into the phe­nom­e­nal break­throughs in Chan­das-śās­tra was com­posed by Piṅ­gala-nāga around 3rd cen­tu­ry BCE. In his chap­ter eight of Chan­das-śās­tra, Piṅ­gala intro­duces some com­bi­na­to­r­i­al tools called pratyayas which can be employed to study the var­i­ous pos­si­ble metres in San­skrit prosody. The algo­rithms pre­sent­ed by him form the ear­li­est exam­ples of use of recur­sion in Indi­an math­e­mat­ics. In the pre­vi­ous edi­tions of Parni­ka, we looked at the algo­rithm giv­en for Prastāra, Saṅkhyā and Naṣṭa. In the cur­rent arti­cle we shall delve more into fur­ther inter­est­ing algo­rithms or pratyayas enun­ci­at­ed by Piṅ­galācārya.

Algorithm#4 ~ Uddiṣṭa

Piṅ­galācārya presents the the fourth of the six pratyayas — Uddiṣṭa for find­ing the row num­ber of a giv­en met­ri­cal pat­tern in the sequence of the n‑syllabled prastāra (ordered sequence of all com­bi­na­tions, as explained in algo­rithm 1 in pre­vi­ous arti­cles) .

The fol­low­ing sūtras cor­re­spond to the pro­ce­dure of deduc­ing the Uddiṣṭa.प्रतिलोमगणं द्विर्लायम्। ततोग्येकं जह्यात्।

(छन्दःशास्त्रम् ८.२६-२७)

This algo­rithm is essen­tial­ly a reverse engi­neer­ing of the pre­vi­ous algo­rithm Naṣṭa, which was enun­ci­at­ed to find the pat­tern giv­en a row num­ber. Now in Uddiṣṭa, the steps are con­cise­ly expressed to reverse the pre­vi­ous process. The steps pre­sent­ed here in these two sūtras imply essen­tial­ly the fol­low­ing:

1. Start with num­ber 1

2. Scan the pat­tern from the right begin­ning with the first L from the right

3. Dou­ble it when an L is encoun­tered

4. Dou­ble and reduce by 1 when a G is encoun­tered

This also is one among the most ele­gant rep­re­sen­ta­tions of the algo­rithm. To illus­trate we shall con­sid­er the fol­low­ing exam­ple: May we employ the Uddiṣṭa algo­rithm to find the row num­ber of the met­ri­cal form “GLLG” in the prastāra (ordered sequence of all com­bi­na­tions) of a 4‑syllabled meter.

1. We start with 1, and scan from right.

2. The right-most pat­tern is a G and hence we skip all Gs, as per the algo­rithm, until we encounter L

3. The next syl­la­ble, scan­ning from right, is L; So we get 1 x 2 = 2

4. Then we find L. So we get 2 x 2 = 4

5. Final­ly we have G. We get (4 x 2) − 1 = 7

Hence the com­bi­na­tion “GLLG” in the ordered list of a 4‑syllabled meter would be the 7th met­ri­cal form in the prastāra. We have already seen that there will be 2 4 total com­bi­na­tions in the ordered list of prastāra and the giv­en com­bi­na­tion is the sev­enth among six­teen pos­si­ble com­bi­na­tions. You are wel­come to try out pat­terns such as “GLLGGLLG” , “GLGLLGLGGG”, and more such pat­terns you would like to play with and write to us ( mail at anaa­di .org ) your answers.

It is indeed enthralling to know in depth about the rich sci­en­tif­ic her­itage of the Indi­an civ­i­liza­tion. We shall con­tin­ue to see oth­er inge­nious achieve­ments in Indi­an math­e­mat­ics in the fol­low­ing edi­tions of Parni­ka.

Aum Tat Sat!