I am trying to represent with spirals the growth of a capital as the Bank pays interest.
The analogy would be that the capital is represented by the radius and the passage of time by the motion of the circle with constant angular speed.
The Archimedes spiral (where radius is growing in arithmetic progression) would be the equivalent of simple interest and the logarithmic spiral would represent compound interest.
Going quantitative, I have doubts. I have tried to adapt the formulas as follows (I am stipulating that one turn is 1 year):
SIMPLE INTEREST (Archimides spiral):
$r(\theta ) = a\theta % MathType!MTEF!2!1!+- % feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacaWGYbGaaiikaiabeI7aXjaacMcacqGH9aqpcaWGHbGaeqiUdeha % aa!3DBF! $
would become
$r(\theta )[{\rm{final capital}}] = {\rm{initial capital + a[yearly interest]}}\frac{\theta }{{2\pi }}{\rm{[fraction or multiple of year elapsed]}} % MathType!MTEF!2!1!+- % feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacaWGYbGaaiikaiabeI7aXjaacMcacaGGBbGaaeOzaiaabMgacaqG % UbGaaeyyaiaabYgacaqGGaGaae4yaiaabggacaqGWbGaaeyAaiaabs % hacaqGHbGaaeiBaiaac2facqGH9aqpcaqGPbGaaeOBaiaabMgacaqG % 0bGaaeyAaiaabggacaqGSbGaaeiiaiaabogacaqGHbGaaeiCaiaabM % gacaqG0bGaaeyyaiaabYgacaqGRaGaaeyyaiaabUfacaqG5bGaaeyz % aiaabggacaqGYbGaaeiBaiaabMhacaqGGaGaaeyAaiaab6gacaqG0b % GaaeyzaiaabkhacaqGLbGaae4CaiaabshacaqGDbWaaSaaaeaacqaH % 4oqCaeaacaaIYaGaeqiWdahaaiaabUfacaqGMbGaaeOCaiaabggaca % qGJbGaaeiDaiaabMgacaqGVbGaaeOBaiaabccacaqGVbGaaeOCaiaa % bccacaqGTbGaaeyDaiaabYgacaqG0bGaaeyAaiaabchacaqGSbGaae % yzaiaabccacaqGVbGaaeOzaiaabccacaqG5bGaaeyzaiaabggacaqG % YbGaaeiiaiaabwgacaqGSbGaaeyyaiaabchacaqGZbGaaeyzaiaabs % gacaqGDbaaaa!8D58! $
COMPOUND INTEREST (Logarithmic spiral):
$r(\theta ) = a{e^{\theta b}} % MathType!MTEF!2!1!+- % feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacaWGYbGaaiikaiabeI7aXjaacMcacqGH9aqpcaWGHbGaamyzamaa % CaaaleqabaGaeqiUdeNaamOyaaaaaaa!3FBD! $
would become
$r(\theta ){\rm{[final capital]}} = a[{\rm{initial capital]}}{e^{b[{\rm{yearly interest rate]}}\frac{\theta }{{2\pi }}[{\rm{fraction or multiple of year elapsed]}}}} % MathType!MTEF!2!1!+- % feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacaWGYbGaaiikaiabeI7aXjaacMcacaqGBbGaaeOzaiaabMgacaqG % UbGaaeyyaiaabYgacaqGGaGaae4yaiaabggacaqGWbGaaeyAaiaabs % hacaqGHbGaaeiBaiaab2facqGH9aqpcaWGHbGaai4waiaabMgacaqG % UbGaaeyAaiaabshacaqGPbGaaeyyaiaabYgacaqGGaGaae4yaiaabg % gacaqGWbGaaeyAaiaabshacaqGHbGaaeiBaiaab2facaWGLbWaaWba % aSqabeaacaWGIbGaai4waiaabMhacaqGLbGaaeyyaiaabkhacaqGSb % GaaeyEaiaabccacaqGPbGaaeOBaiaabshacaqGLbGaaeOCaiaabwga % caqGZbGaaeiDaiaabccacaqGYbGaaeyyaiaabshacaqGLbGaaeyxam % aalaaabaGaeqiUdehabaGaaGOmaiabec8aWbaacaGGBbGaaeOzaiaa % bkhacaqGHbGaae4yaiaabshacaqGPbGaae4Baiaab6gacaqGGaGaae % 4BaiaabkhacaqGGaGaaeyBaiaabwhacaqGSbGaaeiDaiaabMgacaqG % WbGaaeiBaiaabwgacaqGGaGaae4BaiaabAgacaqGGaGaaeyEaiaabw % gacaqGHbGaaeOCaiaabccacaqGLbGaaeiBaiaabggacaqGWbGaae4C % aiaabwgacaqGKbGaaeyxaaaaaaa!94C4! $
Would this adaptation work, is this the right way to do it?