An Anomaly of the Logic of Age Ratios
The Time(s) a Parent Will Be Twice the Age of the Child
or In General the Time(s) An Older Person Will Be Twice the Age of a Younger One or That One Event Will Be a Multiple Time Span Length of Another
Rick Garlikov

This year on Feb. 8, my daughter turned 36.  The previous May 14, I had turned 72.  So on her birthday, in whole number years, I was twice her age, and, of course, conversely she was half my age.  On her birthday part of what I wrote her was: "As I pointed out previously, for some three months, starting on your birthday until mine in May, I will be twice your age for the only times in our lives, and that is cool." 

But it struck me as odd that I would be twice her age in whole number years for only about three months instead of for a year, so I thought about it some more, and realized that was not true -- that there would be another time I was also twice her age -- there will be a time when she is 37 and I am 74, not just the time when she is 36 to my 72.  That would leave a gap when I would be 73.  Now, it really seemed odd to me that I could be twice her age twice, with at least a one-year gap in between those two different times.  So I began to try to figure it out more systematically, and the following is what I came up with and wrote to her:

I was roughly 36.75 years old when you were born.  So to be twice your age, I need to be roughly 73.5 years old when you are roughly 36.75.  You will be 36.75 somewhere around November of 2018, but since we are talking only whole number years for birthdays, you will be 36 and I will be 73 then.  So the double is masked by those whole numbers.  The prior whole number match is 36/72 and the subsequent one is 37/74.  Doing this in whole numbers means that my age always has to be an even number for it to be a double of yours.

In the following table, we are whole number doubles during blue periods and not whole number doubles during red periods.

This shows we are double whole-number-wise from the time we are roughly 36/72.75 until we are roughly 36.25/73, and from the time we are 37.25/74 until 38/.  So the gap is from our being 36.25/73 until we are 37.25/74, a gap of one year between by 73^rd and 74^th birthdays, which is when the fractions of your birthday are from .25 in 2019  to .25 again in 2019.  And it is during that time period that I am 73, which doesn’t allow any whole number you are to be half that.

I can see why we cannot be doubles during the gap, but I am not seeing why we should be able to be doubles on both sides of the gap, other than clearly we are, but I am not seeing the reason that is.  Now if we were both born on January 1, me in 1945, and you in 1982, I would have been 37 when you were born, and then in 2019, the whole year we would be double and half, 74/37.   And it seems to me that, plus intuition in general, means there should always be a whole year when parents and children are double/half in age, but I am not able to see how that means the year gap.  I can see why there is a gap caused by different birthdays during the year, but I am not seeing how there is a double/half on both sides of that gap, to fill out the part of the year that goes with the other gap on the front side of it. 

So then I figured out the following:

Double Age of Parent/Child Explained

Okay, I think I have it now.  This is the general rationale; consider the chart from yesterday (above) with your birthday and mine to be an example:

Explanation A: Rounded off explanation first, just talking about age in years:

1) Whenever the child is born, the parent will already be a particular age, call that age N.

2) In N more years then, on the date of the child’s N^th birthday, the child will become N years old (same age as parent was when child was born).  The parent will already have turned 2N years old and so will be twice the child’s age.

3) The parent will stay twice as old as the child until the parent’s next birthday, which will make the parent then be (2N + 1) years old, while the child is still N years old, so they are no longer twice/half each other’s age.

4) The next birthday will be the child’s, at which time the child will become N + 1 years old.

5) At that time, the parent will still be 2N + 1 years old, and that is not 2(N + 1), so they will still not be double/half each other’s age.

6) The next birthday will be the parent’s birthday (exactly one year after the birthday in step 3), and the parent will then be 2N + 2, which is 2(N + 1), so once again they will be double/half in age until the next birthday, which will be the child’s N + 2 birthday, at which point they will never be twice/half again.

7) Since the date in 6 and the date in 3 are the parent’s successive birthdays exactly one year apart, they will have been not double/half for exactly one year, though they were double/half for the part of the year between the parent’s previous birthday and the child’s N^th birthday.

This explains the one year gap between when they were first double/half until they are double/half again.

Explanation B: More precise explanation, using not just years, but days, considering just non-leap years with the standard 365 days.   Same explanation will work for leap years, using 366 days instead of 365:

1) 1) Whenever the child is born, the parent will already be a particular age, call that age (N years + X days) (X being the number of day since the parent’s most recent birthday).

2) In N more years then, on the date of the child’s N^th birthday, the child will become N years old (same age in years as the parent was when child was born).  The parent will already have turned 2N years old and so will be twice the child’s age, plus the same number of days, X, since the parent’s birthday.  The parent will be (2N years + X days) old on the child’s N^th birthday.

3) The parent will stay twice as old, in whole number years, as the child until the parent’s next birthday, which will make the parent then be (2N + 1) years old exactly, while the child is still N years old (though more precisely, the child will now be [N years + (365 – X) days] old (since the child’s previous birthday was X days after the parent’s, leaving (365 – X) days till the parent’s subsequent birthday, so they are no longer twice/half each other’s age in whole number years.  Thus, the length of time here that they are double/half each other’s age is (365 - X) days.  (In other words: there are always X days from the parent’s birthday to the child’s ensuing birthday because the child was born X days after the parent’s birthday, and in normal, non-leap years then, that will mean there are (365 – X) days each non-leap year from the child’s birthday to the parent’s ensuing birthday.

4) The next birthday will be the child’s (X days after the parent’s), at which time the child will become N + 1 years old.

5) At that time, in whole number years, the parent will still be (2N + 1) years old, and that is not 2(N + 1), so they will still not be double/half each other’s age.

6) The next birthday will be the parent’s birthday (exactly one year after the birthday in step 3), and the parent will then be 2N + 2, which is 2(N + 1), so once again they will be double/half in age in whole number years, until the next birthday, which will be the child’s N + 2 birthday (X days later), at which point they will never be twice/half again. 

7) Since the date in 6 and the date in 3 are the parent’s successive birthdays exactly one year apart, they will have been not double/half for exactly one year, though they were double/half for the part of the year between the parent’s previous birthday and the child’s N^th birthday.  This explains the exact one year gap between when they were first double/half until they are double/half again.

8) Since they are double/half each other’s age in whole number years the first time, for (365 – X) days, and they are double/half each other’s age the second time until child’s (N + 2) birthday, which is X days after the parent turned double the child’s age for the second time, the total time they are double/half each other’s age is [(365 – X) + X] days, which is 365 days or one year exactly,  Hence, the total time they are double/half each other’s age will be one year, with a one year gap in between, from the parents’ birthday in step 3 until the parent’s birthday in step 6.  

9) When the child is born on the parent’s birthday, X = 0, and (365 – X) = 365, so there will then be no whole number year gap and the amount of time they are double/half will still be one year, just a continuous one.

10) Also, when they have the same birthday, the N^th birthday of the child will be at the exact same time as the 2N^th birthday of the parent and they will be exactly double/half when they are double/half in whole number year, but if X > (365/2) – meaning if the time from the parent’s birthday to the child’s birthday is more than half a year, the exact double/half age will fall in the time of the gap year between their whole number double/half ages.  That is because the initial double/half period is between the child’s birthday and the parent’s, so that time period has to be more than half a year (and thus the period between the parent’s birthday and the child’s next birthday, which is X, has to be less than half a year, or the exact double/half date will fall in the gap when there is no double/half relationship in whole years, because it cannot wait until the following whole year later when that date would fall in the second interval of the double/half ratio.

General Observations/Comments:

A) The reason there are two and only two whole number year ages where the parent's age is twice the child's (when the parent and child do not have the same birthday) is that:

1. the parent's 'relevant' birthday -- the one that determines the parent's age in whole number years when the child is born, and thus the number that, when doubled, will be the whole number age the parent is when the child is half its age -- always comes first, when the two birthdays are different, since the parent was already some age in whole number years (in my case 36), based on its most recent birthday before the child was born, and will thus have its relevant birthday first (though not necessarily in the same calendar year -- not when the child is not born in the same calendar year as the parent's 'relevant' birthday) the first time that the parent will be double the age in whole number years that it was when the child was born;
2. the ensuing birthday of the two individuals, after the parent attains the doubled age, will be the child's, which will then make the child half the parent's age in whole number years.
3. The next ensuing birthday of the two individuals will be the parent's, but, as shown above, the parent's age in whole number years will then cease to be double the child's because the parent will have added a whole number year in age, while the child is still the same age the parent was when the child was born. The parent will have gained one whole number year more, while the child has not, thus temporarily ending the doubling of the child's whole number age.
4. The next ensuing birthday will be the child's, adding a whole number year to its age, but since the parent has only added one whole number year to its age, the parent is still not double the child's age in whole number years.
   
5. However, on the next ensuing birthday, which will be the parent's, the parent will then be two (whole number) years older than it was when it first became double the child's whole number age,
   
6. Since at this point the parent will have added two whole number years to the child's one, the parent will again be twice the child's age in whole number years, in my and my older daughter's case going  from 72/36 to 74/37.
7. On the next ensuing birthday of the two individuals will be the parent's, and that will end the double/half ration in whole number years.

B) The age pairings and gap will be different for different children born in other years, and/or for the other parent with any given child if the parents are not the same whole number ages as each other.

C) If we calculate all this in ages in whole number months, instead of whole number years, the above principles will be the same, but the gap will be smaller because the whole numbers will change faster.  E.g., I was 440 whole number months old when my older child was born, so I was 880 months old when she turned 440 months old.  But her birthday is on the 8th of her birth month, February, and mine is on the 14th of mine, May.  She turned 440 months old Oct.8, and I turned 880 months old Sept. 14.  So in the same month she turned 440 months old to my 880, I added another whole number month to my age, making me 881 to her 440, ending the double temporarily.  In the following month, November, on the 8th, she would add a whole number month to make the ratio 881/441, still not double.  But then 6 days later, on November 14,  I added another month to my whole number month age, making the ratio 882/441 and thus double again until her whole month birthday, December 8.  Notice, that still does not make our birth months line up double/half with our actual day of the year we are double/half in age, November 5, but it gets it much closer.  So basically, the unit of measure, if whole numbers are used, not rounded off, determines the length of the gap and where it occurs.  The same would be true if we compared whole number of days we are double/half with the whole number hours in the day, or with the whole number minutes, or with the whole number seconds, or  whole number microseconds, etc. of when our ages are double/half.  The smaller the number used, the smaller the gap and the closer to the actual half/double exact time.

D)  When the child is born within the same calendar year as the parent's 'relevant' birthday -- the one that determines the parent's whole age in whole number years at the birth of the child, the parent's birthday will be first in the calendar year when they first become double/half ages, because that happens on the child's birthday.  But, if like my daughter and my case, the child is born in the subsequent calendar year from the parent's relevant birthday, the child's birthday will always be earlier in the ensuing calendar years.  Her birthday is always in February each year; mine, in May.  But since a whole year is a cyclical event, my relevant age-determining May birthdays come before her relevant age-determining ones.  If calendar years were not numbered with a starting point each January 1 and an ending point each December 31, but were just twelve months repeated cycles, my relevant birthdays would always be before hers, because I had already had a the relevant birthday when she was born.  And the time periods between our birthdays, no matter how large or small, would always be repeated, with, for example, my ('non-relevant') birthday following hers by 95 days (in non-leap years; 96 in leap years), and hers always following mine by 270 days, leap years or not. 

E) Similar arguments and conclusions will apply to other temporal ratios also -- one event's being 3 times older than, 8 times older than, 15 or 100 or 42.35 times older than another.

F) Application of this to physics: The red and blue colors above correspond artificially, because assigned somewhat arbitrarily (except for aesthetics and font readability within them) to the parts of the cycle that are relevant.  But in physics, in the world of nature, ratios involving cycles or cyclical events coincide with or determine actual physical properties.  And if those ratios change depending on the size of the measure being used or the size of smallest observable unit being perceived by the senses or any instrument, then that can change our perception and/or understanding of the phenomenon being observed or studied.  For example, physicists may already take all this into account in their study, understanding, and experimenting with wave cycles and frequencies, but if not, then doing so might make a difference.  The relative starting times and ages of any waves and their temporal components may be significant.    

This is the best I can do with this.  A real mathematician would see an even larger principle and/or have seen this all more readily and one who is also a physicist would probably know to apply it to something actually important and more useful.  I cannot do that, which is why I am not a real mathematician nor a physicist, but just a person able to calculate some things by brute force and persistence to see at least some logic in the calculations.