Cube
While growing up there were a few issues of technical magazines that really captured my imagination. In the basement of the music store where I took flute lessons, there were some old Byte magazines. I remember being captivated by an article about OS/2 and Taligent, asking permission to take the issue home with me. Byte also sparked my interest in NeXT. For better or worse, this had the ultimate result that I was inspired to huddle over my PC for months to write a graphical file manager for the QNX platform. I’m not sure that anyone ever used it, but it’s still available for download.
Often we would go to the Hershey public library’s annual library sale. Several times I scooped up batches of Scientific American issues. The March 1981 issue occupied many hours of my time. In this issue, Douglas Hofstadter’s “Metamagical Themas” column addressed the Rubik’s cube, and I was hooked.
Hofstadter’s article was a brisk introduction to cubology. But instead of giving a quick how-to on solving the cube, he described the underlying concepts and patterns for manipulating the cube. No solutions: only a framework for building a solution piece by piece. But this only whetted my appetite to struggle on to understand what he had to say about group theory. Hofstadter did teach me how to disassemble and reassemble the cube, which was a great help as I stumbled along. (This is not only less destructive than peeling stickers, but is actually much easier.)
Hofstadter introduced a notation for operations on the cube. Each face had a letter: F, B, L, R, U and D for front, back, left, right, up and down. Any operation on the cube could be expressed in these terms, with exponents to indicate repeated turns of the same face (positive exponents for clockwise turns, prime symbol indicating a counter-clockwise turn). So, for example, to go from Start to a simple cross pattern on each face, you could execute F²B²L²R²U²D². Another pretty pattern, which Hofstadter calls “Dots,” can be reached from Start with BF’UD’LR’BF’.
My first excited discovery was a move that swapped two pairs of edge pieces: F²U²F²U²F²U², alternately (F²U²)³. From this one operation I was eventually able to derive all of the operations I needed to solve the cube.
These are not the shortest operators by a long shot, but these are the ones I came up with. I use Hofstadter’s notation for the cubes that are affected by each operator. These rather awkwardly tend to focus on solving the bottom face; I may take the time eventually to rewrite them in terms of the upper face. And thus we come to my motive in writing this post: I can never find my “cheat sheet” when I want it.
- Edge operators
- Double edge pair swappers
- (dr,dl)(df,db) — L’RDF²D²F²D²F²DR’L
- (dr,dl)(df,db) — F²D²F²D²F²D’F²D²F²D²F²D
- (fr,fl)(dr,dl) — F²D²F²D²F²D²
- (fr,fl)(br,bl) — F²D²F²D²F²B²D²B²D²B²
- Double edge flippers
- (dl)+(dr)+ —
F’L’R’B’FD’F’LF’D²F²D²F²D²
F’L’FDF’BRLF’D²F²D²F²D²
- (dl)+(dr)+ —
- Three-cycles of edges
- (dl,df,dr) — L’DLF²D²F²D²F²D²L’D’LF²D²F²D²F²D²
- (df,dl,db) — F²B²DR’LF²RL’U²D’F²B²
- Double edge pair swappers
- Corner operators
- Double corner pair swappers
- (dlf,drb)(drf,dlb) — L’RDF²D²F²D²F²DR’LD²
- (dlf,drb)(drf,dlb) — F²D²F²D²F²D’F²D²F²D²F²D’
- (dlf,drb)(drf,dlb) — L²R²D²L²R²U²DL²R²D²L²R²U²D
- Meson makers (quarkscrews)
- (drf)+(dlb)– —
F’UFU²F’U²FL’RDF²D²F²D²F²DR’LD²
F’U²FU²F’U’FL’RDF²D²F²D²F²DR’LD²
- (drf)+(dlb)– —
- Three-cycles of corners
- (dlf,drb,drf)+- (introduces a quark) —
RUR’BLB’L’RDF²D²F²D²F²DR’LD²
BL’B’L’RDF²D²F²D²F²DR’LD²RU’R’ - (dlf,drb,drf) —
R²DF²D²F²D²F²DR’LD²R’D’R²DUL’UL
RDF²D²F²D²F²DR’LD²L²U’LU’D’R²DL’
- (dlf,drb,drf)+- (introduces a quark) —
- Double corner pair swappers
Hofstadter’s article inspired me to put in the effort to read his fascinating book Goedel, Escher and Bach. It is not an exaggeration, as I do here, to describe it as a romp through a variety of fields. Perhaps second only to my high-school programming mentor, I owe Hofstadter my interests in math and computer science.
If you’re interested in Hofstadter’s article on the cube, you can order a digital edition from Scientific American.
Famine
I’ve appreciated what Doug Wilson has had to say about contemporary food idolatries. His latest post is a little oblique, but good. Pulling out some punchy sound bites (and completely glossing over the necessary caveats):
I believe that some wives, in the way they pursue “healthy” menu choices for their home, are inadvertently trying to teach their husbands and children how to cheat. . . .
A school district in New York just recently dropped the First Lady’s school lunch program because the kids were hungry all the time. What happens in a family where the first lady there has implemented a similar regime and does not have buy-in from her husband and kids? One of the obvious things is that the husband often has the resources to fix things at lunch with a greasy burger, after obtaining a vow from his co-workers to “not tell a soul.”
. . . [A] man should not work to put food on the table, his own table, and then come away from that table hungry. . . . [I]t is crucial that the home not become a place of tight-fisted denial, where wives become the governess of no, instead of the mistress of yes.
I have no idea how common this is. But I want to sidestep that question and take this in a different, but related, direction. There is one marriage, one family, one house and one table that will endure into eternity. These are the archetypes for our marriages, families, houses and tables. So, what kind of table do we believe that Jesus provides for his own bride? And what kind of table is Jesus’s bride setting for him? Is it a famine or a feast?
I’m referring to the Lord’s supper, the new covenant’s one food law and the fulfillment of all old-covenant feasts. We eat this meal together with Jesus at his table in his house. As his bride, we should adorn the table and prepare a kingly feast. And as priests to the king, we have the privilege and responsibility to serve at his request as his royal chefs. There are no more animal sacrifices, of course; we now offer ourselves and the work of our hands, however imperfect, for his evaluation and approval.
Grain and grapes are the Bible’s repeated image of the fruit of the blessed land. Transformed by man’s week-day labor into bread and wine, God uses them again and again to picture the food of the seated and reigning Messiah-king. Because we are seated with him, Jesus gives us a physical taste of his new kingdom; and it is the actual eating and drinking of real bread and wine that is sacramental, rather than merely reflecting on the idea of bread and wine. The Lord’s supper should, as much as possible, convey the greatness, goodness and richness of Jesus’s kingdom. Wherever possible, it is fitting for the church to enjoy the Lord’s supper weekly and to do so lavishly, with rich bread and good wine.
See also: Pig out, Sabbath, The Lord’s table.
Cold-brew coffee
We’ve started making cold-brew coffee to enjoy on weekend afternoons. What they say is true — it has little of the bitterness of normal coffee, so that it tastes almost sweet. Cold-brew coffee very nearly fulfills the promise of coffee’s aroma.
Here’s how we make it:
- Measure the usual amount of coffee grounds you would use and place it in a French press
- Pour room-temperature water over it; use half the amount of water you would use if you were making a hot brew
- Let it steep for twelve hours
- Press (and optionally filter) the coffee and store it in the fridge
- When you’re ready to drink it, add to it an equal portion of either water or milk
Enjoy! It’s worth the inconvenience of having to prepare well in advance.
Patience
Patience is a fruit of the Spirit:
But the fruit of the Spirit is love, joy, peace, patience, . . . — Gal. 5:22
Patience and faith and wisdom and maturity are all bound up together. Consider Abraham’s faith and patience:
By faith Abraham obeyed when he was called to go out to a place that he was to receive as an inheritance. And he went out, not knowing where he was going. By faith he went to live in the land of promise, as in a foreign land, living in tents with Isaac and Jacob, heirs with him of the same promise. For he was looking forward to the city that has foundations, whose designer and builder is God. — Heb. 11:8-10
From the time of God’s promise to Abraham, it was over four hundred years before his descendants were freed from Egyptian slavery. It was a thousand years before Solomon dedicated God’s palace in what was once Melchizedek’s Salem on Mount Moriah. It was another thousand years before Jesus the new Melchizedek inaugurated the New Jerusalem of his church.
Sometimes the patience that the Spirit wants to forge in us is a thousand-year patience.
What would it look like if we were to have a thousand-year patience in our dreams and visions for God’s church?
That is not to say that we should be complacent, or work with any less fervor. It is simply to recognize that God’s kingdom grows like yeast or like a tree. We see this in our own lives, too: while there are great seasons of spring-like growth, over the long haul maturation and glorification is largely a matter of plodding self-sacrificial faithfulness. While the kingdoms of men might rise and fall quickly by the compulsion and cowing of the sword, God’s kingdom flourishes by the nurture (Eph. 5:26) and cutting (Eph. 6:17) of the sword of the Spirit — the word of God.
But this also expands our horizons: what would it look like to dream thousand-year dreams and pray thousand-year prayers for God’s church? If God intends to bless his people to a thousand generations (Ps. 105:8), as his own personal name attests (Ex. 34:6-7), then things might only just be starting to warm up after another thousand years.
See also the future of Jesus.
Unbelievers?
Bob Kauflin interviewed Marty Machowski, the author of the Gospel Story for Kids Sunday-school curriculum, here. There’s a lot to appreciate about what Machowski has to say, but he makes a shocking statement:
Our children, meeting in classrooms during our Sunday worship services represent the largest group of gathered unbelievers across the world.
From time to time I hear parents expressing similar sentiments — “they’re all unregenerate [or heathens],” “we can’t expect that of him; he’s not saved,” or “her problem is just that she needs to be saved.” Overwhelmingly we speak of evangelizing our children rather than discipling them. We wring our hands over the possibility of giving them false assurance, but we are almost entirely unconcerned about the danger of creating millstones of false doubt.
God does not speak of or relate to our children in this way, and it is dangerous for us to do so. It is dangerous because it trains us and our children to doubt and test the promises of God rather than believing and acting upon them. This is how God speaks of our children:
- He addresses them with commands and encouragements as part of the body of his elect “saints” (e.g., Ex. 20:2, 12; Eph. 1:1, 6:1; Col. 1:2, 3:20)
- He requires their presence in worship (Ex. 10:8-11, Ps. 96:7) and feasts (Deut. 16:9-15). He receives their worship (Matt. 21:16) as a potent spiritual warfare to silence his enemies (Ps. 8:2).
- They trusted in him before they were born (Ps. 71:6) and as infants (Ps. 22:9)
- He is their God (Gen. 17:8, Ezek. 37:21-28, etc.)
- He has promised the Holy Spirit to them (Isa. 59:21)
- He regards them as holy (1 Cor. 7:14)
We ought to speak of and think about our children in the same way that God does. This will not leave us complacent, but will instead motivate us to go about the work of parenting rightly, with full confidence in God’s being already at work in them. Instead of leading our children to the way, we will train them in the way (Prov. 22:6). What we once called evangelism must become full-orbed discipleship. Our children need the gospel, but in just the same way we do — to be continually reminded of the promises and goodness and nearness of God and to be growing in repentance and faith.
See also
- Other posts on parenting
- Vern Poythress’s helpful article, Indifferentism and Rigorism
Sabbath
Some commentators suggest that the structure of the middle section of Deuteronomy follows the ten commandments. Moses, having meditated on the law over the course of thirty-eight years in the wilderness, preaches an inspired sermon to Israel reflecting on the greater meaning and application of the law. There is some minor disagreement as to the exact boundaries within this part of Deuteronomy, but one possibility is given by James Jordan in his book, Covenant Sequence in Leviticus and Deuteronomy:
- First commandment: Deut. 6-11
- Second commandment: Deut. 12-13
- Third commandment: Deut. 14:1-21a
- Fourth commandment: Deut. 14:21b-16:17
- Fifth commandment: Deut. 16:18-18:22
- Sixth commandment: Deut. 19:1-22:8
- Seventh commandment: Deut. 22:9-23:14
- Eighth commandment: Deut. 23:15-24:7
- Ninth commandment: Deut. 24:8-25:3
- Tenth commandment: Deut. 25:4-26:19
This is in keeping with other places such as Proverbs and Matthew 5-7, where we see further wisdom drawn from reflection upon the law: Moses, Solomon and Jesus are all inspired commentators on the ten commandments. This also supports the church’s practice of striving to read and apply the commandments with maximum breadth. For example, Calvin writes that “in almost all the commandments, there are elliptical expressions, and that, therefore, any man would make himself ridiculous by attempting to restrict the spirit of the Law to the strict letter of the words.” He concludes that, “thus, the end of the Fifth Commandment is to render honour to [all] those on whom God bestows it” (Book II, Chapter 8, Section 8), since the Bible understands the term “father” quite broadly. In just the same way, the Westminster Shorter Catechism states that the fifth commandment requires us to bestow honor and perform duties “belonging to everyone in their several places and relations, as superiors, inferiors, or equals.” Paul himself seems to make this application of the fifth commandment in Ephesians 6, if we consider all of verses 1-9 to be joined together. And Moses does likewise in Deut. 16:18ff as suggested above.
This observation lends us an interesting bit of help in understanding how the Sabbath commandment can be transfigured in the new covenant from Sabbath to Lord’s day, from last day to first. In the fourth-commandment section (Deut. 14:21b-16:17), Moses mentions three of the seven feasts that God gave to Israel. We see the full list of feasts spelled out in Leviticus 23, beginning with the weekly Sabbath feast and culminating in the feast of booths. The three feasts that Moses lists here in Deuteronomy are the ones that God required to be celebrated at his house. Reading through the entire section, Moses’ application of the fourth commandment establishes the following principles:
- We obey the fourth commandment by bringing a tithe to God’s house
- We obey the fourth commandment by showing generosity and granting rest to others
- We obey the fourth commandment by keeping God’s appointed feasts at his house
These principles help us to understand how Saturday’s Sabbath is transfigured to Sunday’s Lord’s day in the new covenant. God’s house is the gathering of his people before him in worship, and in the new covenant all of the feasts of Leviticus 23 are fulfilled in one feast, the Lord’s supper. Connecting this to Moses’ application of the fourth commandment, we see that the Sabbath itself is fulfilled in the Lord’s supper. Certainly there is much more that needs to be said, but we can say this: when Jesus’s church gathers in his house to celebrate his feast with him and to bring him tribute, there the fourth commandment is being kept.
This also lends support to the practice of weekly communion.
The picture above was painted by my friend, the very talented Jermaine Powell.
Worship is the foundation of righteousness
The first foundation of righteousness undoubtedly is the worship of God. When it is subverted, all the other parts of righteousness, like a building rent asunder, and in ruins, are racked and scattered. What kind of righteousness do you call it, not to commit theft and rapine, if you, in the meantime, with impious sacrilege, rob God of his glory? or not to defile your body with fornication, if you profane his holy name with blasphemy? or not to take away the life of man, if you strive to cut off and destroy the remembrance of God? It is vain, therefore, to talk of righteousness apart from religion. Such righteousness has no more beauty than the trunk of a body deprived of its head. Nor is religion the principal part merely: it is the very soul by which the whole lives and breathes. Without the fear of God, men do not even observe justice and charity among themselves. We say, then, that the worship of God is the beginning and foundation of righteousness; and that wherever it is wanting, any degree of equity, or continence, or temperance, existing among men themselves, is empty and frivolous in the sight of God. We call it the source and soul of righteousness, in as much as men learn to live together temperately, and without injury, when they revere God as the judge of right and wrong. — Calvin, Institutes of the Christian Religion, book 2, chapter 8, section 11
Common disgrace
In a recent post, I argued that Klaas Schilder was wrong in rejecting the notion of common grace. Schilder writes in chapter 18 of his book Christ and Culture:
Certainly, it is true that sin is being “restrained” and that the curse has not been fully poured out upon the world. However, the same thing can be said about the obedience which in Christ Jesus was again permitted to become a gift of God’s free grace and which by the power of Christ’s Spirit also was able to become a gift of this favour. Whoever calls the restraining of the curse “grace” should at least call the “restraining” of the blessing “judgment.”
He goes on to argue that God’s plan both to judge and to save a certain number of men requires as its very precondition the prolonging of time. Thus he says, this patience of God in the case of unbelievers cannot be exercised for the purpose of showing grace, since it is solely for the purpose of showing judgment:
This prolongation and development are no grace. Nor are they curse or condemnation. That is to say, if one wants to use these terms in a serious way. They are the conditio sine qua non of both, the substratum of both.
As I argued earlier, and as D. A. Carson develops in his book, The Difficult Doctrine of the Love of God, the fallacy here is Schilder’s insisting that God cannot be doing two things at once. Intermediately, God is showing genuine love, kindness and patience toward unbelievers. Ultimately, God intends to judge them. The two of these are not inconsistent.
Returning to Schilder’s charge that we should have to speak of a kind of “common judgment,” I think there is actually a sense in which he is right, except we should adopt the term “common injustice” or “common disgrace.” By this I mean that there is a kind of injustice or disgrace when salvation, vindication and deliverance are delayed. With Job, the Psalmist and others, we have a real basis on which to ask God “how long?” However, like Job, we must be willing to accept the answer that God is doing something greater, beyond our understanding, that the vindication we are waiting for is delayed for some greater purpose.
So, then, we have a kind of “common grace” which is the present patience and longsuffering of God toward unbelievers who will one day suffer his wrath; and a kind of “common disgrace,” which is the present suffering of believers for Jesus’s sake, who will one day be completely vindicated in him.
Letter to the editor
I submitted this letter to the editor of the News & Observer:
Dear Editor,
In every city, on any given week, the most important thing that happens is the meeting of King Jesus with his people in his house. The heavens stoop to earth for the king to hold court with his people, eat with them, and send them out as his servants for the life of the world.
This Sunday morning, a triathlon barricaded southern Wake County. Some travelers waited over 90 minutes before turning back. We left early, prepared to expect delays, but not this complete standstill.
Local leaders will answer to constituents for matters such as the interruption of family trips to the farmers’ market on a beautiful spring day. But many families were kept from church, and our leaders will answer to Jesus for the interruption of his worship. Jesus does not wish his people to be detained when he calls them together. I pray that our leaders would follow the example of Darius in Ezra rather than Sanballat and Tobiah in Nehemiah; that they would exert themselves to encourage and not obstruct the assembling of God’s people. Our cities will be blessed by God if they do so.
Respectfully yours,
Scott Moonen
304 Kite Dr.
Fuquay-Varina
Christ and Culture
What follows are some thoughts on Klaas Schilder’s Christ and Culture (PDF).
I had been aware that there was some falling out between Schilder and followers of Kuyper, but wasn’t sure what the nature of the disagreements were. From this book, it seems at a minimum that there was disagreement over the possibility of common grace, and over the notion of presumptive regeneration of covenant children. Steven Wedgeworth provides some additional background information.
I appreciate many of the points that Schilder stresses in this book. He reminds us that the work of individuals must be evaluated with respect to Jesus — our work is pleasing to God only to the degree that we receive things with thanksgiving and offer our work to him in faith and worship. There is a distinctively Christian way to eat a bowl of ice cream, paint a painting or mow the lawn. Or, to put it in other terms, even our working must undergo a sort of death and resurrection if it is to be pleasing to God. And by extension, if we are to speak with Kuyper of “spheres” of life, the church as the center of worship and the center of the Spirit’s out-breaking into the world, holds a central and formative position relative to all other spheres. Schilder offers all this as a criticism of Kuyper; I’m not familiar enough with Kuyper to know how well it sticks, but it is something with which I agree.
There are some areas where I disagree with Schilder. On the petty side, I disagree with some of his application of imagery from Revelation. More significantly, I question his suggestion that Christianity or that Calvinism should be expected to beget a single peculiar style. I don’t think this is a necessary consequence of his principles, although I do think that as Christianity develops in a nation one would expect to see the church and her worship fostering a more mature or “high” style throughout the culture. The kings who bring gifts to Jesus in Psalm 68, Revelation 21, etc. may be attired differently, but they will all be invested with glory of one sort or another.
My most significant area of disagreement is with Schilder’s deduction (primarily in chapter 18, but appearing throughout) that we cannot speak of a “common grace” in the sense that Kuyper and others would have. He deduces this in a very hyper-Calvinistic manner from God’s eternal decrees: since God intends to condemn the reprobate, everything that they enjoy and do, and even the very lengthening of their life, is fitted for the purpose of their destruction and is not fit to be called grace. But Scripture reveals that God is always doing more than one thing at once; it is false to say that, because God intends to condemn someone, that what he is giving them now is not a genuine gift or an expression of of genuine love. As Mark Horne chides:
So the question arises: Did God love Adam and Eve? Were His good gifts to them a revelation of His love for them, or were they snares meant to hurt them? The answer must be that, though God foreordains whatsoever comes to pass, and ultimately causes all things, God’s gifts and offers of future reward are all genuine expressions of a genuine love. It may be difficult to conceive of how this objective revelation in history is to be reconciled with God’s eternal decrees, yet it is perverse to use the decrees to deny that God’s gifts and promises are motivated by love. The fact is, just as without God’s love there is no ground for God’s jealousy, so without God’s good gifts there is no ground for holding ingrates accountable for how they abuse and pervert these gifts. It was Satan’s strategy, after all, to deny that God loved Adam and Eve. If our inferences from God’s decrees put us in Satan’s camp, we need to rethink our position.
We know from passages like Romans 1:21 that unbelievers no less than believers have an obligation to give thanks to God for all they have. More than that, there is a history in the reformed tradition of recognizing that these gifts from God are in fact a spillover from the cross. While this is not saving grace, it is a gift none the less. This explains passages such as 1 Tim. 4:10. Consider Charles Hodge:
Augustinians do not deny that Christ died for all men. What they deny is that he died equally, and with the same design, for all men. He died for all, that He might arrest the immediate execution of the penalty of the law upon the whole of our apostate race; that He might secure for men the innumnerable blessings attending their state on earth, which, in one important sense, is a state of probation; and that He might lay the foundation for the offer of pardon and reconciliation with God, on condition of faith and repentance.
These are the universally admitted consequences of his satisfaction, and therefore they all come within its design. By this dispensation it is rendered manifest to every intelligent mind in heaven and upon earth, and to the finally impenitent themselves, that the perdition of those that perish is their own fault. They will not come to Christ that they may have life. They refuse to have Him to reign over them. He calls but they will not answer. He says, “Him that cometh to me, I will in no wise cast out.” Every human being who does come is saved.
This is what is meant when it is said, or implied in Scripture, that Christ gave Himself as a propitiation, not for our sins only, but for the sins of the whole world. . . .
While I believe Schilder is wrong in saying that we cannot regard the abilities and work of unbelievers as a genuine gift from God, he is right to remind us that no working is neutral with respect to God. So there are multiple layers we must wrestle with — not only the permissibility of our enjoying the work, but also how we are to regard the worker. And yet there are a great many works of unbelievers that we can receive and enjoy with thanksgiving, and even offer to God in worship.
See also: Common grace.
I gotta have my orange juice. 