More and more slowly. That’s how this function increases:

x = x + 1/x

x = 1
1 + 1/1 = 2
2 + 1/2 = 2_1/2
2_1/2 + 1/2_1/2 = 2_9/10
2_9/10 + 1/2_9/10 = 3_71/290
3_71/290 + 1/3_71/290 = 3_150911/272890 = 3.5530103704…
3.5530103704… + 1/3.5530103704… = 3.8344618428…
3.8344618428… + 1/3.8344618428… = 4.0952546322…
4.0952546322… + 1/4.0952546322… = 4.33943969272…
4.3394396927… + 1/4.3394396927… = 4.56988419035…
4.5698841903… + 1/4.5698841903… = 4.78870811637…
4.7887081163… + 1/4.7887081163… = 4.99753270449…
4.9975327044… + 1/4.9975327044… = 5.19763144503…
[…]

But you can tie the function down, as it were, by changing it to this:

x = 1 + 1/x

x = 1
1 + 1/1 = 2
1 + 1/2 = 1_1/2 = 1.5
1 + 1/1_1/2 = 1_2/3 = 1.66666…
1 + 1/1_2/3 = 1_3/5 = 1.6
1 + 1/1_3/5 = 1_5/8 = 1.625
1 + 1/1_5/8 = 1_8/13 = 1.615384615384615384…
1 + 1/1_8/13 = 1_13/21 = 1.619047619047619047619047619…
1 + 1/1_13/21 = 1_21/34 = 1.617647058823529411764705882…
1 + 1/1_21/34 = 1_34/55 = 1.6181818…
1 + 1/1_34/55 = 1_55/89 = 1.617977528089887640449438202…
1 + 1/1_55/89 = 1_89/144 = 1.6180555…
1 + 1/1_89/144 = 1_144/233 = 1.618025751072961373390557940…
1 + 1/1_144/233 = 1_233/377 = 1.618037135278514588859416446…
1 + 1/1_233/377 = 1_377/610 = 1.618032786885245901639344262…
1 + 1/1_377/610 = 1_610/987 = 1.618034447821681864235055724…
1 + 1/1_610/987 = 1_987/1597 = 1.618033813400125234815278648…
1 + 1/1_987/1597 = 1_1597/2584 = 1.618034055727554179566563468…
1 + 1/1_1597/2584 = 1_2584/4181 = 1.618033963166706529538387946…
1 + 1/1_2584/4181 = 1_4181/6765 = 1.618033998521803399852180340…
1 + 1/1_4181/6765 = 1_6765/10946 = 1.618033985017357938973140873…
[…]

You could call the function “x = 1 + 1/x” a tie-phi, because it generates the golden ratio, a fascinating mathematical constant also known as phi or φ = 1.6180339887498948482… Note how the denominator (the lower part) of each fraction above becomes the numerator (the upper part) of the next fraction. The function is reproducing the Fibonacci sequence, which starts with “1, 1” and proceeds by adding the two previous numbers:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597…

Dividing each number by the previous number yields a better and better approximation of φ (but never reaches the exact value of φ, which is an irrational number, that is, it cannot be expressed as a ratio of two whole numbers). And what about the function x = 1 – 1/x? If you remember that subtracting a negative number is the same as adding a positive number, you can work it out like this:

x = 1 – 1/x

x = 2

1 – 1/2 = 1/2 = 0.5
1 – 1/1/2 = -1
1 – 1/-1 = 2
1 – 1/2 = 1/2 = 0.5
1 – 1/1/2 = -1
1 – 1/-1 = 2
1 – 1/2 = 1/2 = 0.5
1 – 1/1/2 = -1
1 – 1/-1 = 2
[…]

x = 3

1 – 1/3 = 2/3 = 0.66666…
1 – 1/2/3 = -1/2 = -0.5
1 – 1/-1/2 = 3
1 – 1/3 = 2/3 = 0.66666…
1 – 1/2/3 = -1/2 = -0.5
1 – 1/-1/2 = 3
1 – 1/3 = 2/3 = 0.66666…
1 – 1/2/3 = -1/2 = -0.5
1 – 1/-1/2 = 3 = 3
1 – 1/3 = 2/3 = 0.66666…
[…]

x = 73

1 – 1/73 = 72/73 = 0.98630136986301369863013…
1 – 1/72/73 = -1/72 = -0.01388888…
1 – 1/-1/72 = 73
1 – 1/73 = 72/73 = 0.98630136986301369863013…
1 – 1/72/73 = -1/72 = -0.01388888…
1 – 1/-1/72 = 73
1 – 1/73 = 72/73 = 0.98630136986301369863013…
1 – 1/72/73 = -1/72 = -0.01388888…
1 – 1/-1/72 = 73
[…]

The function falls into an endless loop of period 3. But what about using the two functions together, first x = 1 + 1/x, then x = 1 – 1/x, then repeating?

loop(x = 1 + 1/x, x = 1 – 1/x)

x = 1
1 + 1/1 = 2
1 – 1/2 = 1/2 = 0.5
1 + 1/1/2 = 3
1 – 1/3 = 2/3 = 0.666666…
1 + 1/2/3 = 2_1/2 = 2.5
1 – 1/2_1/2 = 3/5 = 0.6
1 + 1/3/5 = 2_2/3 = 2.66666…
1 – 1/2_2/3 = 5/8 = 0.625
1 + 1/5/8 = 2_3/5 = 2.6
1 – 1/2_3/5 = 8/13 = 0.615384615384…
1 + 1/8/13 = 2_5/8 = 2.625
1 – 1/2_5/8 = 13/21 = 0.619047619047619…
1 + 1/13/21 = 2_8/13 = 2.615384615384615384…
1 – 1/2_8/13 = 21/34 = 0.6176470588235294117647058824…
1 + 1/21/34 = 2_13/21 = 2.619047619047619047619047619…
1 – 1/2_13/21 = 34/55 = 0.6181818181818181818181818182…
1 + 1/34/55 = 2_21/34 = 2.617647058823529411764705882…
1 – 1/2_21/34 = 55/89 = 0.6179775280898876404494382022…
1 + 1/55/89 = 2_34/55 = 2.6181818…
1 – 1/2_34/55 = 89/144 = 0.618055555…
1 + 1/89/144 = 2_55/89 = 2.617977528089887640449438202…
1 – 1/2_55/89 = 144/233 = 0.6180257510729613733905579399…
1 + 1/144/233 = 2_89/144 = 2.618055555…
1 – 1/2_89/144 = 233/377 = 0.6180371352785145888594164456…
1 + 1/233/377 = 2_144/233 = 2.618025751072961373390557940…
1 – 1/2_144/233 = 377/610 = 0.6180327868852459016393442623…
1 + 1/377/610 = 2_233/377 = 2.618037135278514588859416446…
1 – 1/2_233/377 = 610/987 = 0.6180344478216818642350557244…
1 + 1/610/987 = 2_377/610 = 2.618032786885245901639344262…
1 – 1/2_377/610 = 987/1597 = 0.6180338134001252348152786475…
1 + 1/987/1597 = 2_610/987 = 2.618034447821681864235055724…
1 – 1/2_610/987 = 1597/2584 = 0.6180340557275541795665634675…
1 + 1/1597/2584 = 2_987/1597 = 2.618033813400125234815278648…
1 – 1/2_987/1597 = 2584/4181 = 0.6180339631667065295383879455…
1 + 1/2584/4181 = 2_1597/2584 = 2.618034055727554179566563468…
1 – 1/2_1597/2584 = 4181/6765 = 0.61803399852180339985218034…
1 + 1/0_4181/6765 = 2_2584/4181 = 2.618033963166706529538387946…
1 – 1/2_2584/4181 = 6765/10946 = 0.6180339850173579389731408734…
1 + 1/0_6765/10946 = 2_4181/6765 = 2.61803399852180339985218034…
1 – 1/2_4181/6765 = 10946/17711 = 0.6180339901755970865563773926…

Phi and the Fibonacci sequence are back, but in a zig-zagging or two-steps-forward, one-step-back kind of way. Now adapt the two previous functions slightly. Here’s the plus function adapted:

x = 2 + 1/x

x = 1

2 + 1/1 = 3…
2 + 1/3 = 2_1/3 = 2.33333…
2 + 1/2_1/3 = 2_3/7 = 2.428571428571428571…
2 + 1/2_3/7 = 2_7/17 = 2.411764705882352941176470588…
2 + 1/2_7/17 = 2_17/41 = 2.414634146341463…
2 + 1/2_17/41 = 2_41/99 = 2.41414141…
2 + 1/2_41/99 = 2_99/239 = 2.414225941422594142259414226…
2 + 1/2_99/239 = 2_239/577 = 2.414211438474870017331022530…
2 + 1/2_239/577 = 2_577/1393 = 2.414213926776740847092605887…
2 + 1/2_577/1393 = 2_1393/3363 = 2.414213499851323223312518585…
2 + 1/2_1393/3363 = 2_3363/8119 = 2.414213573100135484665599212…
2 + 1/2_3363/8119 = 2_8119/19601 = 2.414213560532625886434365594…
2 + 1/2_8119/19601 = 2_19601/47321 = 2.414213562688869635045751358…
2 + 1/2_19601/47321 = 2_47321/114243 = 2.414213562318916695114799156…
2 + 1/2_47321/114243 = 2_114243/275807 = 2.414213562382390584720474825…
[…]

This function is generating Pell numbers and approximating the value 1 + √2, where √2 = 1.414213562373095048801688724… Now try adapting the minus function:

x = 2 – 1/x

x = 2

2 – 1/2 = 1_1/2 = 1.5
2 – 1/1_1/2 = 1_1/3 = 1.33333…
2 – 1/1_1/3 = 1_1/4 = 1.25
2 – 1/1_1/4 = 1_1/5 = 1.2
2 – 1/1_1/5 = 1_1/6 = 1.166666…
2 – 1/1_1/6 = 1_1/7 = 1.142857142857142857…
2 – 1/1_1/7 = 1_1/8 = 1.125
2 – 1/1_1/8 = 1_1/9 = 1.11111…
2 – 1/1_1/9 = 1_1/10 = 1.1
2 – 1/1_1/10 = 1_1/11 = 1.09090909…
2 – 1/1_1/11 = 1_1/12 = 1.0833333…
2 – 1/1_1/12 = 1_1/13 = 1.0769230769230769230…
2 – 1/1_1/13 = 1_1/14 = 1.0714285714285714285…
2 – 1/1_1/14 = 1_1/15 = 1.0666666…
2 – 1/1_1/15 = 1_1/16 = 1.0625
[…]

Where x = 1 – 1/x falls into an endless loop, x = 2 – 1/x endlessly falls towards 1. Now try combining x = 2 + 1/x and x = 2 – 1/x. The result might be surprising:

loop(x = 2 + 1/x, x = 2 – 1/x)

x = 1

2 + 1/1 = 3
2 – 1/3 = 1_2/3 = 1.66666…
2 + 1/1_2/3 = 2_3/5 = 2.6
2 – 1/2_3/5 = 1_8/13 = 1.615384615384615384…
2 + 1/1_8/13 = 2_13/21 = 2.619047619047619047619047619…
2 – 1/2_13/21 = 1_34/55 = 1.618181818181818181818181818…
2 + 1/1_34/55 = 2_55/89 = 2.617977528089887640449438202…
2 – 1/2_55/89 = 1_144/233 = 1.618025751072961373390557940…
2 + 1/1_144/233 = 2_233/377 = 2.618037135278514588859416446…
2 – 1/2_233/377 = 1_610/987 = 1.618034447821681864235055724…
2 + 1/1_610/987 = 2_987/1597 = 2.618033813400125234815278648…
2 – 1/2_987/1597 = 1_2584/4181 = 1.618033963166706529538387946…
2 + 1/1_2584/4181 = 2_4181/6765 = 2.618033998521803399852180340…
2 – 1/2_4181/6765 = 1_10946/17711 = 1.618033990175597086556377393…
2 + 1/1_10946/17711 = 2_17711/28657 = 2.618033988205325051470844820…
2 – 1/2_17711/28657 = 1_46368/75025 = 1.618033988670443185604798401…
2 + 1/1_46368/75025 = 2_75025/121393 = 2.618033988780242682856507377…
2 – 1/2_75025/121393 = 1_196418/317811 = 1.618033988754322537608830406…
2 + 1/1_196418/317811 = 2_317811/514229 = 2.618033988748203621343798191…
2 – 1/2_317811/514229 = 1_832040/1346269 = 1.618033988749648101530971893…
2 + 1/1_832040/1346269 = 2_1346269/2178309 = 2.618033988749989097047296779…
2 – 1/2_1346269/2178309 = 1_3524578/5702887 = 1.618033988749908598925421458…
2 + 1/1_3524578/5702887 = 2_5702887/9227465 = 2.618033988749889595896597820…
2 – 1/2_5702887/9227465 = 1_14930352/24157817 = 1.618033988749894081903178586…
2 + 1/1_14930352/24157817 = 2_24157817/39088169 = 2.618033988749895140905679158…
2 – 1/2_24157817/39088169 = 1_63245986/102334155 = 1.618033988749894890909100681…
2 + 1/1_63245986/102334155 = 2_102334155/165580141 = 2.618033988749894831892914018…
2 – 1/2_102334155/165580141 = 1_267914296/433494437 = 1.618033988749894845824745843…
2 + 1/1_267914296/433494437 = 2_433494437/701408733 = 2.618033988749894849113605205…
2 – 1/2_433494437/701408733 = 1_1134903170/1836311903 = 1.618033988749894848337210827…
2 + 1/1_1134903170/1836311903 = 2_1836311903/2971215073 = 2.618033988749894848153928977…
2 – 1/2_1836311903/2971215073 = 1_4807526976/7778742049 = 1.618033988749894848197195953…
2 + 1/1_4807526976/7778742049 = 2_7778742049/12586269025 = 2.618033988749894848207409900…
2 – 1/2_7778742049/12586269025 = 1_20365011074/32951280099 = 1.618033988749894848204998714…
2 + 1/1_20365011074/32951280099 = 2_32951280099/53316291173 = 2.618033988749894848204429510…
2 – 1/2_32951280099/53316291173 = 1_86267571272/139583862445 = 1.618033988749894848204563881…
2 + 1/1_86267571272/139583862445 = 2_139583862445/225851433717 = 2.618033988749894848204595602…
2 – 1/2_139583862445/225851433717 = 1_365435296162/591286729879 = 1.618033988749894848204588114…
2 + 1/1_365435296162/591286729879 = 2_591286729879/956722026041 = 2.618033988749894848204586346…
2 – 1/2_591286729879/956722026041 = 1_1548008755920/2504730781961 = 1.618033988749894848204586763…
[…]

It’s a tie-phi again, with better and better aproximations to φ and φ + 1 = φ^2. Now another function that may be another surprise:

x = 5 – 5/x

x = 2

5 – 5/2 = 2_1/2 = 2.5
5 – 5/2_1/2 = 3 = 3
5 – 5/3 = 3_1/3 = 3.333333…
5 – 5/3_1/3 = 3_1/2 = 3.5
5 – 5/3_1/2 = 3_4/7 = 3.571428571428571428…
5 – 5/3_4/7 = 3_3/5 = 3.6
5 – 5/3_3/5 = 3_11/18 = 3.611111…
5 – 5/3_11/18 = 3_8/13 = 3.615384615384615384615384615…
5 – 5/3_8/13 = 3_29/47 = 3.61702127659574468085106383…
5 – 5/3_29/47 = 3_21/34 = 3.617647058823529411764705882…
5 – 5/3_21/34 = 3_76/123 = 3.617886178861788617886178862…
5 – 5/3_76/123 = 3_55/89 = 3.617977528089887640449438202…
5 – 5/3_55/89 = 3_199/322 = 3.618012422360248447204968944…
5 – 5/3_199/322 = 3_144/233 = 3.61802575107296137339055794…
5 – 5/3_144/233 = 3_521/843 = 3.618030842230130486358244365…
5 – 5/3_521/843 = 3_377/610 = 3.618032786885245901639344262…
5 – 5/3_377/610 = 3_1364/2207 = 3.618033529678296329859537834…
5 – 5/3_1364/2207 = 3_987/1597 = 3.618033813400125234815278648…
5 – 5/3_987/1597 = 3_3571/5778 = 3.618033921772239529248875043…
5 – 5/3_3571/5778 = 3_2584/4181 = 3.618033963166706529538387946…
5 – 5/3_2584/4181 = 3_9349/15127 = 3.618033978977986381966021022…
5 – 5/3_9349/15127 = 3_6765/10946 = 3.618033985017357938973140873…
5 – 5/3_6765/10946 = 3_24476/39603 = 3.618033987324192611670833018…
5 – 5/3_24476/39603 = 3_17711/28657 = 3.61803398820532505147084482…
5 – 5/3_17711/28657 = 3_64079/103682 = 3.618033988541887695067610579…
[…]

It’s a tie-phi with the Fibonacci sequence again. But only the Fibonacci sequence. The Lucas sequence too:

1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207, 3571, 5778…

The Lucas sequence is generated by the same rule as the Fibonacci sequence — add the two previous numbers — but it’s seeded with “2, 1” rather than “1, 1”.

Finally, if you’er wondering where cy-phi is, it’s here:

x = 1 – 1/x

x = φ

φ – 1/φ = 2 – φ = 0.3819660112501052…
2-φ – 1/(2-φ) = -φ = -1.6180339887498948…
-φ – 1/-φ = φ = 1.6180339887498948…
2-φ = 0.3819660112501052…
-φ = -1.6180339887498948…
φ 1.6180339887498948…
2-φ = 0.3819660112501052…
-φ = -1.6180339887498948…
[…]

That’s a cycling phi or cy-phi.